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# Set Theory - Science topic

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This is a question about Godel Numbering. As I understand it, the axioms of a system are mapped to a set of composite numbers. Is this really the case, so for example the 5 axioms of Euclidean plane geometry are mapped to 5 composite numbers? Does this also imply that theorems of the system are now composite numbers that are dependent on the composite numbers that were the target of the map from the set of axioms PLUS the elementary numbers that describe the logical operations, such as +, if..then, There exists, ext.?
From what I understand, you are asking whether it is sensible to investigate the nature of numbers coding axioms (finitely many) of a given theory, or more generally, study a formal theory with respect to its coding and see whether it is possible to "extrapolate" from theorems provable in this theory a number representing a code of a formulation of an axiom. It has already been said that the Gödel numbering used in the proof of the Incompleteness theorem is canonical and probably given its computational inefficiency maybe also inadequate for this particular question (again, assuming that I understood your question correctly).
However, the main idea of understanding how theorems are, in some sense, computationally linked to the premises which entail them is a very interesting question. It is clear that the possibility of arithmetizing a formal theory enables to provide a number theoretical interpretation to the objects of a formal theory (syntax, semantics and proof theory). In this sense, one could ask whether it is possible to find a "preferable" coding for a formal theory in order to make your question easier to formulate and answer. This is of course extremely vague, but to make things more precise, you could try to pick a specific theory (Euclidean Geometry), formalise the axioms and define different codings and see whether, with respect to what you are trying to investigate, it is possible to establish how a particular choice of coding affects the nature of the numbers coding your axioms.
Finally, I wanted to conclude by referring to a recent program in mathematical logic called Reverse Mathematics. They are basically trying to isolate axioms from theorems (which is some sense related to the remark you made "[..]going in the other direction, starting with a result in number theory and then try to conclude something about formal systems or axioms.") and they actually use a lot of computability theory and subsystems of second-order arithmetic. I am not an expert on this particular topic, but I suggest you to look into it if you are interested.
I would be happy to carry on this discussion with you and also, if during the past year you obtained some interesting results that you can share, I would be glad to hear them.
Thank you and best regards!
Jean Paul Schemeil
Simpson, Stephen G. (2009), Subsystems of second-order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, doi:10.1017/CBO9780511581007, ISBN 978-0-521-88439-6, MR 2517689
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Dear all,
I have been finding several published papers on soft set, soft topology and hybrid structures. In early days of my career in soft topology, I also worked with notions of Cagman et al., Shabir and Naz, etc. But, I must admit boldly that ideas of soft set theory, soft topology, etc. defined by others are incorrect. Prof. Molodtsov raised questions many times on available notions of soft empty set, soft absolute set, etc. He showed disagreements on these notions which carry no sense of soft set in reality. Prof. Molodtsov had disagreements on operations of soft sets, soft mappings, etc. His arguments and disagreements were published in several journals, but I am sad to say that soft set researchers community never tried to find Prof. Molodtsov, the father of soft set, along with his works on soft sets.
Thus, I am bold enough to ask following questions to researchers of soft topology:
1) How can you claim that you are following correct notions and meanings of soft topology?
2) Have you ever tried enough to search about Prof. Molodtsov and his published papers post 1999's soft set theory before publishing rubbish ideas in spite of the fact that Prof. Molodtsov showed us correct path in soft set, soft topology, soft rational analysis, etc.? The published papers of others are not contributing any correct idea to soft set theory and hybrid structures by anyway. These papers are simply misleading young researchers because of some arrogant researchers of soft set theory who are desperately ready to reject ideas and advices of Prof. Molodtsov, the father of soft set theory just for publication bof their ideas or to increase citations.
Thus, I request editors of journals to stop publishing incorrect ideas of soft set theory. I also request publishing houses to encourage correct notions and ideas of soft set theory and hybrid structures. I also request American Mathematical Society, European Mathematical Society, etc. to take initiative to organise conferences on soft set theory for the discussion of correct structures of soft set, soft topology, etc. I am available for debates on correct structures of soft set theory and related ideas in international conferences if one is ready to debate. I just want that soft set theory and hybrid structures must be developed using correct ideas of Prof. Molodtsov.
Thanks
Santanu Acharjee, PhD
(Collaborator of Prof. D. Molodtsov,
Title of the paper: Soft rational line integrals, published in 2021 and two more in submitted form)
Assistant Professor
Department of Mathematics
Gauhati University, India
Dear Dr. Fawaz,
Kindly don't share me the chapters with incorrect ideas on soft set theory. Read my questions at first very carefully. Publication of a paper on soft set doesn't mean that the notions, definitions, etc. of soft set used in the paper are correct.
Don't be blind to accept the truth that Prof. Molodtsov raised several questions on this incorrect notions and he published several paper.
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Edit: I updated question 1 and added a link in Section 3.1. I also explained the intuition in section 4.4.1.
I have yet to understand amenability and group theory, but the questions in my attachment might be of interest.
I would be glad if all the questions were answered; however, if you wish for a particular question, read section 4 of my paper and the question in section 5.
I want an elegant choice function where, for specific A in 4.4, gives the structure (defined in 4.2) that I'm looking for.
follow
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How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?
Regards,
Germán Benitez Monsalve
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Minimal requirements for standard model of set theory leading to inconsistency?
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There Is No Standard Model of ZFC and ZFC_2 with Henkin Semantics
An posible generalization of the Löb's theorem. Jaykov Foukzon*, Israel Institute of Technology, Haifa, Israel (1089-03-60)
Attention: the topic is littered with stupid amateur comments
Gold words!
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Re: ARTICLE: "Should Type Theory replace Set Theory as the Foundation of Mathematics?" BY
Thorsten Altenkirch
Type Theory is indicated (by the author) to be a sometimes better alternative and a sometimes-replacement for regular set theory AND thus a sometimes better replacement for the logical foundations for math (and Science). It seems to allow turning what is qualitative and not amenable to regular set theory into things that can be the clear particular objects of logical reasoning. Is this the case? (<-- REALLY, I am asking you.)
It is very rarely, if ever, I have addressed anything that I did not have a good understanding of; BUT, here is the exception (and a BIG one). (I HAVE VERY, VERY little understanding of this Article -- even from the most crude qualitative standpoint. You would say I should have researched this more, but it in not my bailiwick , only more confusion, on my part would likely occur, "shedding no light". My sincere apologies. ANYHOW:
:
If indeed things are as the author, Thorsten Altenkirch, says: it seems different things (other than those related to standard propositions in regular set theory) could widen the use of set theory itself yet retaining (including) all of regular set theory (with all of its virtues, as needed). BUT, in addition it is indicated it could be applied to areas (PERHAPS, like biological and behavior science) where present set theory (and the math founded on it) cannot now be applied.
"[ The ] type theoretic axiom of choice hardly corresponds to the axiom of choice as it is used in set theory. Indeed, it is not an axiom but just a derivable fact."
More Quoting of the author: "Mathematicians would normally avoid non-structural properties, because they entail that results are may not be transferable between different representations of the same concept. However, frequently non-structural properties are exploited to prove structural properties and then it is not clear whether the result is transferable." .... "And because we cannot talk about elements in isolation it is not possible to even state non-structural properties of the natural numbers. Indeed, we cannot distinguish different representations, for example using binary numbers instead." ... "we can actually play the same trick as in set theory and define our number classes as subsets of the largest number class we want to consider and we have indeed the subset relations we may expect. ... Hence Type Theory allows us to do basically the same things as set theory" ... as far as numbers are concerned (modulo the question of constructivity) but in a more disciplined fashion limiting the statements we can express and prove to purely structural ones."
"we cannot talk about elements in isolation. This means that we cannot observe intensional properties of our constructions. This already applies to Intensional Type Theory, so for example we cannot observe any difference between two functions which are pointwise equal." ...
"...Hence in ITT (regular set theory) while we cannot distinguish extensionally equal functions we do not identify them either. This seems to be a rather inconvenient incomplete- ness of ITT, [ (common set theory)] which is overcome by Type Theory (HoTT)"
"[It] reflects mathematical practice to view isomorphic structures as equal. However, this is certainly not supported by set theory which can distinguish isomorphic structures. Yes, indeed all structural properties are preserved but what exactly are those. In HoTT all properties are structural, hence the problem disappears. ..."
"While not all developments can be done constructively it is worthwhile to know the difference and the difference shouldn’t be relegated to prose but should be a mathematical statement." [AND}: ...
"Mathematicians think and they often implicitly assume that isomorphic representations are interchangeable, which at closer inspection isn’t correct when working in set theory. Modern Type Theory goes one step further by stating that isomorphic representations are actually equal, indeed because they are always interchangeable."...
..."The two main features that distinguish set theory and type theory: con- structive reasoning and univalence are not independent of each other. Indeed by being more explicit about choices we have made we can frequently avoid using the axiom of choice which is used to resurrect choices hidden in a proposition. Replacing propositions by types shows that that the axiom of choice in many cases is only needed because conventional logic limits us to think about propositions when we should have used more general types."
Oh, here's the link to THE ARTICLE:
The answer is simply no. Additionally, considering "realist (platonic)" and "non-realist (non-platonic)" doesn't actually help with the answer I am going to provide, and the article also begs the question. It's like asking why you like music, is it because it sounds good, or is it because it makes you feel good? Well, that depends on what you mean! Equally, asking a working mathematician about the independence of math, or the construction of math will get you very confused looks. They way one treats math, is ever which is the most convenient, or the most sensible to the person. As such, the article in question does not particularly respect nor delineate the historical and functional differences between these two foundations of mathematics very well. Mathematics is a very broad, messy, overlapping subject. In fact, most of the math I regularly use does not really involve calculations, or functions per se. But, as the article is a pre-print, I assume it simply represents a scribbling of his thoughts.
In order to elucidate my answer better, some background in the cartography of mathematics is needed. There are many different universes (formal distinct foundations of mathematics as unique fields) of mathematics that have their own level of reasoning, and focuses. To name a few, category theory, abstract group theory, analysis, proof theory, many-valued logic, and the list just keeps going. All of which are employed at different levels to ascertain certain properties of math, or even to articulate certain questions. For instance, if one wants to study the different universes of mathematics, category theory is generally involved, and the object considered is called a topos. Or if one wishes to study how numbers work, one can employ number theory to study them as unique things, or you can employ analysis and study them as functions, or you can study them with group theory and consider them as action as well. In this view, no field of mathematics has a primacy over other mathematics, only advantages to the inquiries at hand.
Here is a simple question that I think illustrates the point I am making: is two an element of four, or not? That is to ask, in the construction of numbers, are they considered logically unique (aka type theory), or as informal primitives so that numbers are just simply numbers (set theory)? It is in fact this very question that helps separate type theory and set theory. This question, is akin to asking is meaning found in words or what the words represent? However both are true to a certain degree, and from different perspectives. If we are partial to the former, we are essentially asking, does the construction of words form the meaning they express? Yes, but only if we consider meaning as inherent to language alone (intensional). That is language makes meaning, not the world outside of our minds. If we are partial to the latter however, then words denote things, they are analogs to events, and point to common descriptions that we see (extensional). In the same manner, type theory considers numbers as things in themselves, to say "there are two dogs" is to say two dogs. Because the number two is different then dogs. Equally, computer scientists often employ type theory because it logically constructs things, whereas, mathematicians like set theory because its very good at describing things, and there relationships. It would be very burdensome for a mathematician if we had to logically construct everything from the bottom up. Instead of saying, let us consider a sequence of integers. The computer scientist would have to define every part of that sentence.
I hope this helps clarify the question.
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I have digged into a theory established by D. Loeb from University of Bordeaux, and found an interesting way to represent the things you have (classical set) and the things you do not have and might need, want, be ready to acquire (new forms of set with negative number of elements)
This fits very well in the extension of the matrices of sets which I needed to develop, se why and how here (see references below).
I would be thrilled to know if you have use cases where this model of classical sets of what you have, and new negative sets of what you do not have, my help.
REF:
[2]
Further to Peter's answer, another way of putting it would be to say that a set with a positive or nil number of elements can exist on its own, whereas a set with a negative number of elements can only be a virtual set that must exist in relation or combination with other more 'normal' sets, and may be engaged in operations affecting those sets. It cannot exist independently, but can take on a transient role as part of an operation or operator on those sets, or intermediate results.
By the way, if I may ? It's 'I dug into a theory..', not 'I have digged ...'
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Considering a matrix A which has vectors v1=[1;0;0] and v2=[1;1;0] i.e. this matrix A is spanned by the vectors v1 and v2.
The rank of this matrix A is 2. Going by the definition of the rank of a matrix it means the number of independent vectors or the dimension of the row space.
Seeing A={v1,v2} with a cardinality of 2 an we say that the cardinality is the same as the rank of the matrix which in turn means that it gives the number of independent vectors spanning A
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows.
By definition, two sets are of the same cardinality if there exists a one-to-one correspondence between their elements. For a finite set, the cardinality is the number of its elements. ... For example, Z and R are infinite sets of different cardinalities while Z and Q are infinite sets of the same cardinality.
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Hi all,
Can the gen keyword be used to run stepwise calculations for a molecule using lower to higher order basis sets/theory? What I was not sure about is let's say I start with pm6->b3lyp->m06, will the chk file be updated and used progressively to make calculations easier for the next level of theory?
Thanks for any input!
That makes a lot of sense with the link command, thank you very much for your reply Massimiliano Arca !
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I run into this problem all the time. We have physical samples with a number of properties that are being measured. Those parameters are not necessarily normally distributed. We want to split the entirety of these samples into subsets for experiments. We want the subsets to be of equal distribution in each dimension, as far as possible that is.
Random sampling is not useful since it mostly generates subsets which are worse than hand picking. At the moment most of us do hand pick or just run random samples.
I mostly iterate random sampling and use a Kolmogorov-Smirnov test in each dimension to check the 'quality' of the subsets against each other and the entirety of the data. I don't know how to express this problem mathematically or programmatically to run a proper optimization of subsets but I have the feeling this can be solved much more elegantly and effectively. To me it feels like a reversed latin-hypercube problem. Minimize correlation and maximize space filling.
I'm grateful for any ideas, readily available functions or papers/books related to it.
Andreas
Have a look at https://m-py.github.io/anticlust/, a package that accompanies the paper .
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Soft set was introduced soft set in 1999 as a model to capture uncertainties. Recently, Smarandache introduced a notion which he calls hyper soft set. In soft set theory a soft is associated with a function which assigns a subset of the universe of discourse to each parameter.
In case of hyper soft sets, also we have a function which associates every parametric value tuple with a subset of the universe of discourse.
As claimed by Smarandache, his notion generalises an earlier notion of Gamma soft sets. That is when number of parameters is 2 we gat Gamma soft sets.
But, as dealt by the authors of Gama soft set, the second component is a fixed set of parameters. This does not seem to be the case of hyper soft sets.
So, my query is why the notion of Smarandache be called as hyper soft set. It would have been termed so if we are getting soft set when the number of parameters is reduced to 1.
I would like to get inputs from researchers in soft set theory as well as hyper soft set theory.
Yes we can Soft set is the special case of Hypersoft Set.
The definition of Hypersoft Set is mapping say F,
F: A_1×A_2×A_3....×A_n to P(U)
Here each A_i is further classified.
Where's in case of Soft Set A_i are not further categorize.
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We have seen a stability in the supply chains of goods, food in particular, during the current pandemic of Covid19 continue, mostly undisturbed.
It is very reassuring at a time of uncertainty and macro-risks falling onto societies.
How much do we owe to the optimised management and supervision of Container transport, and multimodal support to it with deep sea vessels, harbour feeder vessels, trains and trucks/lorries?
What is the granularity involved? Hub to hub, regional distribution, local delivery?
Do we think that the connectivity models with matrices, modelling the transport connections, the flows per category (passengers, freight, within freight: categories of goods), could benefit from a synthetic model agreggation of a single matrix of set federating what has been so far spread over several separate matrices of numbers?
What do you think?
Below references on container transport, and on matrices of sets
REF
A) Matrices of set
[i] a simple rationale
[ii] use for containers
[iii] tutorial
B) Containers
[1] Generating scenarios for simulation and optimization of container terminal logistics by Sönke Hartmann, 2002
[2] Optimising Container Placement in a Sea Harbour, PhD thesis by by Yachba Khedidja
[3] Impact of integrating the intelligent product concept into the container supply chain platform, PhD thesis by Mohamed Yassine Samiri
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Do you think the "New Whole Numbers Classification" exactly describes the organization of set N ?
Whole numbers are subdivided into these two categories:
- ultimates: an ultimate number not admits any non-trivial divisor (whole number) being less than it.
- non-ultimates: a non-ultimate number admits at least one non-trivial divisor (whole number) being less than it.
Non-ultimate numbers are subdivided into these two categories:
- raiseds: a raised number is a non-ultimate number, power of an ultimate number.
- composites: a composite number is a non-ultimate and not raised number admitting at least two different divisors.
Composite numbers are subdivided into these two categories:
- pure composites: a pure composite number is a non-ultimate and not raised number admitting no raised number as divisor.
- mixed composites: a mixed composite number is a non-ultimate and not raised number admitting at least a raised number as divisor.
From the paper:
very interesting....
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Is vague set is more applicable then fuzzy set?
It has been established that vague sets are intuitionistic fuzzy sets in an article:
H.Bustince and P.Burillo: Vague sets are intuitionistic fuzzy sets, Fuzzy sets and systems, 79(3), (1996), pp. 403-405.
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I can't find any manual or procedure on how to conduct the MMDE process. It looks like most of the literature has copy-pasted the stuff without any method of how to do to it. Any practical book, paper or support is much appreciated. I don't have much expertise in mathematics or set theories that's why not able to apprehend the concepts.
Thank you so much Muhammad Ali
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How does one get access to the Mizar Mathematical Library (MML) ? This refers
to the Mizar system for the formalisation and automatic checking of mathematical proofs based
on Tarski-Grothendieck Set Theory (mizar.org).
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Self-contained sets are accepted in set theory. An example is the set E of all sets having more than two members, which has more than two members itself; hence E∈ E. Likewise, most subsets of E have also more than two members; therefore most subsets of E are also members of E. This statement can be written as follows
P_2(E) ⊆ E. (1)
where P_2(E) denotes the collection of all subsets of E having "more than two members".
Likewise, denote by K(E) the collection of all subsets of E having "at most two members." It is a straightforward consequence of these definitions that
P(E) = P_2(E) ∪ K(E) (2)
where P(E) stands for the powerset of E.
Now suppose that E is an infinite countable set. It is not difficult to see, that the collection K(E) is also countable; accordingly, since P(E) is non-countable, by virtue of the equation (2), P_2(E) must be non-countable too. Thus, the equation (2) means that the countable set E contains a non-countable subset P_2(E).
This seems like a variant of Barber-Russell paradox.
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suppose in 2 dimensional space there are two triangles. Now there are different cases as follows
1> they have no intersection
2> they intersect on a point
3> they intersect on an edge
4> they intersect on a 2D area
I want to quantify these intersections using a measure defined on sets. Is there a measure that can be helpful in this pursuit.
Zubair
What is wrong with the usual (Lebesgue) measure?
It gives zero for the first and second cases. And length for the third case.
The area for the last case.
It is not clear where is the unusual situation and the difficulty of the problem.
Best regards
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As The First Generation of Infinite Set Theory is based on present classical infinite theory system, contradictory concepts of "potential infinite” and “actual infinite" make people unable to understand at all what the mathematical things being quantitative cognized in set theory are-------- are they "potential infinite things” or “actual infinite things " or the mixtures of both or none of both? People have been unable to understand at all what kind of relationship between the quantitative cognizing theory and the unavoidable concepts of "potential infinite, actual infinite" in set theory: If the mathematical things being quantitative cognized are "potential infinite”, what kind of "potential infinite” cognizing idea, operations and results should people have; if the mathematical things being quantitative cognized are "actual infinite”, what kind of "actual infinite” cognizing idea, operations and results should people have; if the mathematical things being quantitative cognized are the mixtures of both or none of both "potential infinite” and "actual infinite”, what kind of mixing cognizing idea, operations and results should people have? Are there "one to one correspondence" theories and operations for "potential infinite elements” or “actual infinite elements" or the mixtures of both or none of both? Why?
Therefore, it is very free and arbitrary for people to conduct quantitative cognitions to any infinite related mathematical things in The First Generation of Infinite Set Theory: It can either be proved that there are as many elements in Rational Number Set as there are in Natural Number Set or that there are more elements in Rational Number Set than that in Natural Number Set; the T = {x|x📷x}theory can either be used to create Russell’s Paradox or to create "Power Set Theorem", make up the story of “the Hilbert Hotel forever with available rooms” ------- strictly make all the family members of the Russell's Paradox mathematicization and turn all the family members of Russell's Paradox into all kinds of Russell's Theorems; ...
However, because it has a little to do with applied mathematics; it is impossible to verify the scientificity of many practical quantitative cognitive operations and results in set theory. So, there are far more unscientific contents (more arbitrary quantitative cognizing behaviors) in the quantitative cognitive process of present classical infinite set theory than in present classical mathematical analysis.
You have raised many simultaneous questions about the (finite) and (infinite) concepts. Indeed the latest (infinite) is full of paradoxes in mathematics.
One can observe that :
(*) the part may include the whole,
(*) many infinite sums rearranged to obtain different answers,
(*) ambiguity of Cantor sets,
(*) infinity is not real,
All terms: inf - inf, 0xinf, inf/inf, inf^inf, inf^0 1^inf all are undefined!!
So one can stay with such paradoxes years without any clear answer.
And this doesn't mean that we can't use infinity. It is useful to find particular answers for a given mathematical problem. Also, we can construct new definitions that should be consistent with the axioms of the set theory and all other branches of current modern mathematics. All are considered valid based on the added axiom. This is very similar to change the fifth postulate of Euclid's to construct hundreds of non-euclidean geometries; all are consistent and accepted.
So, you can say that the sum of the angles of the triangle is 180 degrees or > 180degree, or < 180 degrees all are correct but in different geometries.
All agree with the initial axioms, but they differ by one axiom.
We can do the same for the set theory.
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The emergence of the new infinite theory system determines the emergence of the second generation of infinite set theory and the fourth generation of mathematical analysis------- we call present classical infinite set theory basing on present classical infinite theory system “the first generation of infinite set theory (the old infinite set theory)" and the new infinite set theory basing on the new infinite theory system the second generation of infinite set theory"; we call "the first, second and third generations of mathematical analysis (before standard analysis, standard analysis and non-standard analysis) " basing on present classical infinite theory system “the present classical mathematical analysis (the old mathematical analysis) " and the new mathematical analysis basing on the new infinite theory system the fourth generation of mathematical analysis".
The advantage of the potentially infinite, to achieve the infinite by some limit of the finite, is that there would not be this abrupt break with properties of finite sets.
So finally it seems to me there can be two radically different approaches to infinite sets.
This is consistent with the view of many analysts, who say infinity as a number does not exist, potential infinity is just as large as needed in each step. This is the view of intuitionists also. Somehow this sets a limit to methods of induction.
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We have understood from the studies of infinite related mathematics’ history that present classical infinite theory system is based on the concepts of "potential infinite and actual infinite", which cannot be defined scientifically and contradict each other. This fatal flaw in the basic theory deeply affects the scientific nature of mathematical behavior of mathematical workers in the field related to the concept of "infinite". So, one cannot escape the constraint of the two false concepts of "potential infinite and actual infinite", and one cannot stop the emergence of various infinite related paradoxes. In addition, these paradoxes must exist in the form of "family (infinite paradox syndrome)". In different historical periods, the constantly emerging paradox family members repeatedly reveal the fundamental defects in the classical infinite theory system from different perspectives and call on people to solve these very defects. The fatal fundamental defects in present classical infinite theory system are the source of the second and the third mathematical crisis: more than 2500 years, no one can get rid of a kind of disease in the infinite related fields of mathematics --------- a diseases produced by the confusion of "potential infinite and actual infinite" concepts in set theory diagnosed clearly by Poincare, Frege, and Weyl more than 100 years ago. Studies have proved that this is the common "disease" existing in many infinite related mathematical disciplines with the foundation of present classical infinite theory system: the various "number and non--number mathematical things” -------- “variables of not only potential infinite but also actual infinite (the ‘ghost’ disappearing and reappearing at any time?)" for all the family members of Zeno's Paradox and Berkeley's Paradox in mathematical analysis [1-6]; the mathematical things with the property of "elements belonging and not belonging to a set ---------- T = {x|x📷x} (variable elements of not only potential infinite but also actual infinite: the ‘ghost’ disappearing and reappearing at any time?) " for all the family members of Russell paradox in set theory；…
This is why we are so sure to say that the Third Mathematical Crisis in present scientific theory system is unsolvable and the Third Mathematical Crisis is another manifestation of the Second Mathematical Crisis in set theory. They are "twins".
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In the "second generation of infinite set theory" based on the new infinite theory system, the "infinite carrier theory" is the basic theory for quantitative cognitions and studies of many "mathematical things related to infinite" in infinite set. The generation and existence of the new mathematical carrier “interim-set, Non-Definite-Set 📷” determines the generation and existence of the new system of set types (Set Spectrum) --------- Empty Set 📷; Finite-Set 📷; Interim-Set (Non-Definite-Set) 📷; Infinite Set 📷, Yan Set 📷. In this newly constructed infinite related Set Spectrum: "zero quantity" is the only quantitative cognizing result of the “Empty Set 📷” since there are no elements contained in such set, which determines that people do not need or can not conduct any more quantitative cognitions and studies on the number of elements contained in such set.; many different finite sets contain different number of elements since Finite-Set 📷 since the elements contained in such sets are the “finite numbers with cognizable quantitative nature (Archimedean Property)" and people must conduct quantitative cognitions and various studies on the number of elements contained in such sets; people are bound to conduct quantitative cognitions and various studies on the number of elements contained in Interim-Set (Non-Definite-Set) 📷 since the elements contained in such sets are the “infinite law carriers with cognizable quantitative properties (Half Archimedean Property)”.
You are using unusual terms to describe the set theory:
(first generation, second generation, etc.). What is the philosophy behind your classification?
In the mathematical language, one needs to show and define a set of consistent axioms to construct a new model. And next, compare the old model with the new one. I think that it is challenging to add any new axiom for the current model.
To talk about the advantages and some new numerical analysis techniques is a different issue. Where the global vision can be analyzed locally using different approaches, but all should be performed under the same umbrella of the main mother set theory.
Best regards
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I need to know areas of artificial intelligence with uncertainty in which soft set theory or its other generalizations can be applied.
There are many applications that use fuzzy logic in artificial intelligence techniques to solve their problems, including:
Medical diagnostics
Image processing and signal: from image compression, image enhancements, encryption, image watermarking , etc.
Intrusion detection of networks
and etc...... Good Luck
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I have a very good parsimonious solution with 4 solution terms with appropriate consistency and coverage values. However, the intermediate solution has only 3 solution terms i.e. some parsimonious terms overlap making it hard to interpret.
Can I ignore the intermediate solution and interpret only the parsimonious solution?
Are there any good reference papers that can elaborate and justify the decision?
Thank You
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this is axiomatic set theory . these axioms are needed for set theory and not for mathematics. so can we avoid them since the involve use of predicate and property. will experts guide in detail. can the use be restricted by using a mapping rather than property or predicate notion ?
You mentioned in your question that "these axioms are needed for set theory and not for mathematics", but this is not a simple claim.
Separation and replacement axioms are needed for establishing many important results dear to mathematicians. For instance, it is fundamentally necessary to prove recursion theorem for natural numbers. Replacement, on the other hand, is necessary for establishing transfinite recursion.
(Note: one do not need separation, for it is provable using replacement)
One can use functions directly in replacement axiom.
Replacement: if F is a function and A any set, then {F(x) | x in A} is a set.
However, the stronger version would be provable using the other axioms of ZF and the weak version of replacement.
Strong replacement:
If F(x, y) is a property the behaves like a function, then for any A, {F(x) | x in A} is a set.
********
In the chapter Constructible Sets of Set Theory: Third Millenium Edition (Thomas Jech), Jech defines Godel operations for building the contructible universe. This can be seen as a strategy for considering the generation of sets as operations. This may interest you.
About your question: I keep hearing that some subtheory of "hereditarily finite" set theory is OK that way, but I have only a fuzzy idea what it is and am too lazy to look it up ...
Hereditarily finite sets axiomatization is known as the 'set theory equivalent of peano arithmetics'. These theories are very closely connected: they are bi-interpretable. The idea is: you remove the infinity axiom, say that every set is finite and that every member of each set is finite. In this theory, the axiom of separation and replacement become the equivalent of the axiom of induction in arithmetics.
Notably, it is known that ZF can provide a model construction for PA and thus it can provide a model construction for this set theory. Using completeness theory for first order logic, it means that ZF proves consistency of this theory.
But, this is even more general. ZF has a property called reflection. It means that ZF provides truth predicates for any set-size part of itself. In particular, the class of hereditarily finite sets is a set in ZF. Therefore, ZF provides a truth predicate for it, i.e. ZF proves the consistency of hereditarily finite set theory.
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As "the first generation of infinite set theory" is based on present classical infinite theory system, contradictory concepts of "potential infinite” and “actual infinite" make people unable to understand at all what the mathematical things being quantitative cognized in set theory are-------- are they "potential infinite things” or “actual infinite things " or the mixtures of both or none of both? People have been unable to understand at all what kind of relationship between the quantitative cognizing theory and the unavoidable concepts of "potential infinite, actual infinite" in set theory: If the mathematical things being quantitative cognized are "potential infinite”, what kind of "potential infinite” cognizing idea, operations and results should people have; if the mathematical things being quantitative cognized are "actual infinite”, what kind of "actual infinite” cognizing idea, operations and results should people have; if the mathematical things being quantitative cognized are the mixtures of both or none of both "potential infinite” and "actual infinite”, what kind of mixing cognizing idea, operations and results should people have? Is there "one to one correspondence" theories and operations for "potential infinite elements” or “actual infinite elements" or the mixtures of both or none of both? Why?
As it turns out, the quantitative cognizing theories and operations (including the theory and operations of one to one correspondence and limit theory) for those infinite related mathematical things in "the first generation of infinite set theory" are lack of scientific foundations: It is impossible to know at all what the relationship among all the quantitative cognizing behaviors in infinite set theory and the concepts of "potential infinite” and “actual infinite" is and how to carry out scientific and effective operations specifically to different kinds of infinite related mathematical things. Therefore, it is very free and arbitrary for people to conduct quantitative cognitions to any infinite related mathematical things in "the first generation of infinite set theory" : It can either be proved that there are as many elements in Rational Number Set as there are in Natural Number Set or that there are more elements in Rational Number Set than there are in Natural Number Set; the T = {x|x📷x}theory can either be used to create the Russell’s Paradox or to create "Power Set Theorem", make up the story of “the Hilbert Hotel forever with available rooms” ------- strictly make all the family members of the Russell's Paradox mathematicization and turn all the family members of Russell's Paradox into all kinds of Russell's Theorem; ... However, because it has nothing to do with applied mathematics, it is impossible to verify the scientificity of many practical quantitative cognitive operations and results in set theory. Therefore, there are far more unscientific contents in the quantitative cognitive process of present classical infinite set theory than in present classical mathematical analysis, because it can be more arbitrary！
Dear Geng Ouyang,
Read word by word, paragraph by paragraph, chapter by chapter the book "Set Theory and it's Logic" by Willard Van Orman Quine, and you will understand my reaction to your writings. Good luck!
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Because "the first generation of infinite set theory" (Cantor’s set theory) is based on present classical infinite theory system, it is bound to be unable to get rid of the confusion of "potential infinite and actual infinite" contents, and people have to take the quantity of elements in an infinite set as the mixed number forms of "potential infinite number and actual infinite number" derived from those concepts of "potential infinite and actual infinite". This inevitably leads to two contradictory cognitive behaviors: On the one hand people deny the necessary relationship between "element” and “set" (not knowing at all that the existence of "elements with different properties" leads to the existence of different infinite sets?), firmly deny the characteristic differences of elements contained in different infinite sets (the unique existing meaning, unique existing form and unique existing condition as well as unique relationship between), and construct a kind of “cardinal number theory” by "double abstraction" which is conflict with the nature of infinite set. Through the "double abstraction", elements contained in many infinite sets are turned into piles of “geometric points" without any differences of “unique existing meaning, unique existing form and unique existing condition as well as unique relationship between”, to ensure that they all have the same "cardinal number" (to ensure many different infinite sets have the same quantity of elements). After "double abstraction", elements contained in different infinite sets have lost their original unique properties (including the property of number) and become "endless infinite geometric point"--------"endless" becomes the only numerical property for all the elements contained in many infinite sets (as we know, all the points on the lines are just piles of "endless abstract things" without any differences of “unique existing meaning, unique existing form, unique numerical property and unique existing condition as well as unique relationship between”. All the points on the line segments are with the same "cardinal number" and heir quantities are uniformly "infinite"). So, in present classical set theory (Cantor’s set theory), after "double abstraction", many subsets and their original sets contain same amount of elements -------- the elements contained in many infinite sets have the same "cardinal number" and "endless" becomes the only numerical property for all the elements contained in many infinite sets. On the other hand, all in a sudden, people recognized the necessary relationship between "elements and sets", firmly recognized the importance of the essential differences in the manifestation, nature, existing conditions and relationship among the elements in infinite sets, recognized that it is the existence of "elements with different characteristics" that leads to the existence of different infinite sets; all in a sudden, people denied the "double abstraction theory”, suddenly decided not to apply "double abstraction theory” in the cognitive process for elements contained in the infinite set, so as to ensure the applying of T = {x|x📷x}theory which has nothing to do with "double abstracted” to find some elements still with their special original features (not being "double abstracted”) and to complete some proofs that some infinite sets contain more infinite elements than other sets (for example, the infinite elements contained in Real Number Set are more than the infinite elements contained in Natural Number Set, the infinite elements contained in any infinite set are less than the infinite elements contained in its power set -------- Infinite Real Number Set is more infinite than Infinite Natural Set Number Set, and any infinite set is less infinite than its power set,... For hundreds of years, people have been trying so hard to study and fabricate various "infinite concepts", various formal logic, formal languages and "assembly line" operations related to those "contradictory and colorful concepts of infinite". However, the fundamental defects revealed by these two problems have determined the impossibility of scientific, effective and systematic qualitative and quantitative studies to elements contained in infinite sets, paradoxes are inevitably produced -------- because it is impossible to know at all what the infinite related elements contained in infinite sets are (the abstract things that are both potential infinite and actual infinite: the "ghost" disappearing and reappearing at any time?). Therefore, we draw an important conclusion: "the third mathematical crisis" is another manifestation of "the second mathematical crisis" in set theory. They are "twins". Studies have shown that, the unavoidable conceptual confusion of "potential infinite, actual infinite" in present classical infinite theory system determines Cantor's theory and operations of “cardinal number” and "double abstraction" are not self-justification at all and lack of scientific and systematic, which inevitably results in the impossibility of scientific, effective and systematic qualitative and quantitative studies to elements contained in infinite sets.
Nothing wrong. All correct. Wrong are your attempts to find errors in this theory. Get busy, buddy!
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The musical melody is a structure consisting of a series of two types of entities: tones and pauses. Each tone has two properties: pitch and duration; each pause has one property - duration. According to these properties, they can compare to each other. The result of a comparison can be identity or difference.
Hypothesis: some combination of tones and pauses give us a sense of beauty, others don’t. Let us assume that beauty is proportional to the quantity and variety of the identity relations that the melody structure contains.[1]
Question: how can we determine the quantity and variety of identity relations in a given melody structure if we know that there are:
1. identity relations between individual tones and pauses;
2. identity relations between relations. (example: A and B are different in the same (identical) way as B and C; duration of A is half of the duration of B just like (identically) the duration of C is half of the duration of D; etc.)[2]
3. between groups of tones (and pauses)
And a second question: by which method can we create structures that contain maximum quantity and variety of identity relations?
*********
[1] About the reasons behind this hypothesis seePreprint , part 3.
[2] The structure must be observed throw time. If we play the tones and pauses of a beautiful melody in random time order the beauty will be lost. These types of relations allow us that.
Quite true. There is a danger that rationalization of beauty can lead to its destruction. Explaining a joke just makes it not funny. However, curiosity, desire for knowledge, seems to be stronger.
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I reject Cantor's infinite set theory as inconsistent (see The Countable-Infinity Contradiction ). Is mathematics more than the unbounded set of consistent definitions formed from the concept of 1?
I mentioned in my previous answer the need of the dimension concept to build the geometry; this includes the points which are defined as objects that have zero dimension. To locate the position of the point, we need the algebraic system of numbers. For example, p = p(x, y, z) ∈ R3 where x, y, and z are numbers represents the length, width, and height.
Best regards
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There are two confusions when i apply QCA in my research.
1. how to deal with omitted variables? To my knowledge, most of QCA papers incorporate about 5 to 7 conditions. Too many conditions often make the results difficult to be interpreted. Then, some variables have been omitted. traditional regression analysis also suffer from the problem of omitted variables, but we can add more control variables to the model so as to reduce this risk.
2. what is the adequate level of solution coverage? In my research, the solution coverage scores are 0.55 and 0.7. Some reviewers argue that the coverage are too low to make rigorous claims. They take an analogy of R squares in regression analysis.
Thank you for sharing your ideas.
Dear Yimin
1. When designing your QCA research, the best (if imperfect) way of trying to address the problem is to base your selection of conditions as much as possible on existing theoretical and substantive knowledge, so as to ensure you have included potentially relevant conditions. You can also combine several conditions into higher-order constructs, in order to keep the number of conditions at a manageable level (once you have conceptualized these conditions, this can be implemented with the compute() function in the QCA package based on calibrated component sets).
In case-oriented variants of QCA, a good way of identifying problems with omitted variable is via in-depth study of cases. On the one hand, cases that are not covered by the solution can tell you about alternative pathways to the outcome. On the other hand, in-depth study of typical cases and deviant cases for consistency can tell you about whether the conditions you included in the truth table actually provide a good explanation, or whether there is something you left out. I find the various works by Rohlfing and Schneider on set-theoretic multi-method research very useful in this regard. It can all be implemented via the SetMethods package by Nena Oana and Schneider.
Such insights can then feed back into a redefinition of the relevant explanatory framework in order to be able to explain more cases--that is part of QCA as an iterative approach.
Data mining might enable you to maximize coverage and consistency, it might not guarantee that the identified models adequately explain the cases and reflect relevant concepts and theories--these are eventually interpretations to be made using criteria external to the QCA algorithm and parameters of fit, such as theory, case knowledge, and additional methods of analysis. But these are more general issues in empirical social research, not really peculiar to QCA. A middle way between in-depth study of cases and simple data-mining can be to try out several equally plausible and meaningful options (for example, in terms of conditions considered, or in terms of options for conceptualizing, measuring, and/or calibrating the sets) considering the state of theoretical and substantive knowledge, and then compare the resulting models regarding their performance.
2. The coverage measure can be interpreted as % cases covered by the solution for crisp sets. It cannot be interpreted as such for fuzzy sets, so the analogy has clear limits there. But of course, it is about how much of the variation in the outcome we could describe.
Low coverage means that there is a large share of your empirics that is described or explained by other conditions than the one you have considered. So, it might be a good motivator to go back to point 1 and try and think of additional or different conditions to consider. There may be situations when you are not interested in covering as many cases as possible, but some particular cases of theoretical relevance to you--in which case coverage might be less of an issue. Still, as a researcher I would want to know why I could not cover the other cases, so my analysis would not stop there. There is also an approach to QCA that requires solution coverage of at least 0.75 (see the work by Michael Baumgartner), in order for a solution to be causally interpretable. But most applied QCA currently does not feature concrete thresholds for coverage sufficiency. As a general principle, I would say that the lower the solution coverage, the more our solution is mainly to be interpreted as a descriptive statement of the complex configurations that characterize the cases covered; and the less it can be interpreted more generally as the conditions that typically imply the outcome under question. We've demonstrably missed out on a lot of those.
I hope this helps, all best,
Eva
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Algorithm
VennPainter uses set-theory to generate Venn diagrams. The intersection is defined as follow:
A \ B ¼ fx : x 2 A ^ x 2 Bg
and its complement:
B n A ¼ B \ A∁ ¼ fx : x 2 B ^ x 2= Ag
Technically, integer ax is assigned to label element x.
ax can represent the following: 8x 2 ax ¼ i¼ [n Xn Ai; 1 ≤ n ≤ 31 bi; 1 ≤ n ≤ 31 and bi ¼ i ¼ 1 i ( 2i—1; x 2 A 0; x 2= Ai
Thus, if ax1 = ax2, then x1 and x2 belong to the same intersection. VennPainter labels every intersection Um with an integer cUm in the Venn diagram (S1, S2 and S3 Figs). If ax ¼ cUm , then x 2 Um. The ﬂowchart (Fig 3) shows how VennPainter works.
These authors contributed equally to this work.
not sure here what to explain in that algorithm...
So, let me try to explain why ax1 = 21.
Well, obviously x1 belongs to A1, and that is why we have b1 = 2^0 = 1;
Next, x1 does not belongs to A2, and that is why we have b2 = 0;
Next, x1 belongs to A3, and that is why we have b3 = 2^2 = 4;
Next, x1 does not belongs to A4, and that is why we have b4 = 0;
Finally, x1 belongs to A5, and that is why we have b5 = 2^4 = 16.
By definition, we have ax1 = b1 + b2 + b3 + b4 + b5 = 21.
Kind regards,
Anton
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Recently, I am very interested in neutrosophic set & theory. In my best knowledge, a neutrosophic set can be represented by three quantities T, I, F (the truth, the indeterminacy and the falsity). But, in some recent literature, I have read that a neutrosophic number can be represented in the form z=a+bI, where a, b are real numbers and the quantity "I" denotes for the indeterminacy. I cannot find out the relation between two concepts. Could you please give me some details and suggestions for this problem ? Sincerely,
See the following reference
[Studies in Fuzziness and Soft Computing] Fuzzy Multi-criteria Decision-Making Using Neutrosophic Sets Volume 369 || Interval-Valued Neutrosophic Numbers with WASPAS
Kahraman, Cengiz, Otay, İrem
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According to the results of Gödel [1940] and Cohen [1963, 1964] it is accepted fact that CH is independent of axiomatic set theory. In plain terms, an analysts using ZF can assume that CH is either true or false and no argument can be made to contradict the assumed axiom.
Given that the question of P vs. NP must be predicated on some general asymptotic model of computation, it is natural to assume that some transfinite well-ordered model from ZF/NGB be the most appropriate way to attack the problem. In this type of analysis, one would have to show that the cardinality of a solution is related to the solution or state space size of the solution. Then one simply needs to look at the asymptotic solution behavior as a function of a linear data input size.
Under such a strategy, NP contains as least the lowest exponential (2^n) cardinality aleph 1, and arguably P contains the smallest cardinality aleph 0 (n).
If we assume that CH holds, then there is no cardinality c between n < c < 2^n. This implies that:
• · P != NP because there is a clear bifurcation between aleph 1 and aleph 0 and
• · L=P (where L are all linear solutions) because there are no intermediate solution cardinalities c in P except for those that reduce to L.
On the other hand, if we assume !CH holds then there exists c between n < c < 2^n which bifurcates P from NP implying P !=NP.
According to the independence of CH from ZF either is equally valid but as a outlined above both would lead to the same result that P != NP. However the assumption that CH holds implies that all problems in P have solution L. This leads us to assume that !CH because certainly not all problems in P are actually L?
We have a philosophical quandary here. Do we truly believe that CH is independent of our best mathematical analytic methods? Given the strong relation to P vs. NP are we also to believe that P vs. NP is also independent of ZF/NGB? Even if P vs. NP is independent of ZF we find that we must still conclude that P !=NP.
Perhaps it is ZF/NBG that is truly bifurcated from computational reality and therefore independent?
· Cohen, Paul J. (December 15, 1963). "The Independence of the Continuum Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. 50 (6): 1143–1148. Bibcode:1963PNAS...50.1143C. doi:10.1073/pnas.50.6.1143. JSTOR 71858. PMC 221287. PMID 16578557.
· Cohen, Paul J. (January 15, 1964). "The Independence of the Continuum Hypothesis, II". Proceedings of the National Academy of Sciences of the United States of America. 51 (1): 105–110. Bibcode:1964PNAS...51..105C. doi:10.1073/pnas.51.1.105. JSTOR 72252. PMC 300611. PMID 16591132.
· Gödel, K. (1940). The Consistency of the Continuum-Hypothesis. Princeton University Press.
I am inclined to consider that CH is irrelevant with regard to P vs NP, although Choice might be a concern somehow.
I say that with respect to CH because the ordinary models of computation are pretty much confined to the denumerable and rational terminating interpretations of reals. That functions (and all the predicates) are not denumerable, along with undecideability, does not require a commitment with respect to CH as far as I can tell.
I hope it is that simple.
With regard to P vs NP, there are odd situations in practice with regard to how some presumably NP hard worst cases can still yield to useful heuristics. I think it is this sort of thing that leads Knuth to surmise that eventually (in the limit?) P = NP will arrive. I worry that he's swimming against the tide, but ... .
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Hey policy and/or social scientists,
I am trying to analyze 3 conditions for an outcome, my N is relatively small with 6 countries I am trying to compare.
I have generated a few necessary conditions so far, but for some reason the standard analysis generates only a parsimonious solution with three paths. They all have a consistency of .7 or larger each and the solution coverage as well as consistency is .667. HOWEVER, none of the solution paths is represented in the truth table, so none of the cases fulfills one of the paths.
I have attached a screenshot of the Truth Table. Hope someone can give me some clarity! Is my N simply too small?
Dear Victor,
fs/QCA 2.5/3.0 often spits out "ERROR (Quine-McCluskey): The 1 Matrix is Empty" or "ERROR(Quine-McCluskey): The 1 Matrix Contains All Configurations".
The first error message is returned when all your output values are negative, i.e. when there is not a single row in the truth table which is accepted as being sufficient for the outcome. This happens when your consistency cut-off is higher than the consistency score of the best-performing row).
The seond error message is the opposite. It is returned when all your output values are positive, i.e. when there is not a single row in the truth table which is accepted as not being sufficient for the outcome. This happens when your consistency cut-off is lower than the consistency score of the worst-performing row).
Neither case is good of course. For QCA to work, you need at least one row that is negative and one row that is positive.
However, you will also notice that even though all of your rows are positive / negative, fs/QCA will still produce complex and intermediate solutions. This is only possible because, for producing these two solution types, fs/QCA adds artifical data to your data set through an algorithmic back door (if you want to know more about this process, let me know, it would lead too far here). In other words, these two solution types enlarge your data set without you noticing, and with some quite desastrous consequences (it can be shown that both solution types not only produce inferences beyond your data, but that these inferences are often leading you way off path).
I hope this helps.
Best wishes,
Alrik
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The unavoidable fatal defect of “’potential infinite--actual infinite’ confusion” in present classical infinite idea inevitably leads to the unceasingly production of “paradox events” (different in forms but same in nature) from many infinite relating fields in present science theory system and, the self-contradictory (Self-refutation Mechanism) “self and non-self” contents in present set theory (such as T={x|x📷x}) and mathematical analysis (such as the number-of-non-number variable) is a typical example. It is true that people have been trying very hard to solve those infinite relating paradoxes, but the mistaken working idea brought very little effect-------since antiquity, people have been unaware of that these suspended “infinite paradox events” are in fact an “infinite paradox syndrome” disclosing from different angles exactly the same fundamental defects in present classical infinite theory system, have not been studying seriously the consanguineous ties among the paradoxes in the syndrome, have not been studying seriously the consanguineous relations among these paradoxes and the foundations of their related theory systems (such as number system) , have not been studying seriously and deeply the fundamental defects in present classical infinite theory system disclosed jointly by different infinite paradox families; but merely studied, made up and developed very hard all kinds of formal languages, formal operations and formal logics specially for solving surface problems. So, not only these “infinite paradox families” have never been solved but developing and expanding unceasingly.
v
Thank you dear Mr. Dennis Hamilton!
According to my studies, Zeno's great creation of “Achilles--Tortoise paradox” is not only a simple mistake but is a huge paradox family and its typical modern family member is the newly discovered Harmonic Series Paradox.
Let’s see following divergent proof of Harmonic Series which can still be found in many current higher mathematical books written in all kinds of languages:
1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．． （1）
=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．． （２） >1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．． （3） =1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity （4）
Because of not knowing what infinitesimals are, the unavoidable practical problem has been troubling us ever since is how many items (including infinitesimals of cause) in infinite decreasing Harmonic Series can be added up by “brackets-placing rule" to produce infinite numbers each bigger than 1/2?
This kind of “infinite-infinitesimals paradox” tells us:
1, in Harmonic Series, we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite infinitesimals in Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity;
2, the “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items in Harmonic Series corresponds to different runners with different speed in Zeno’s Paradox while the items in Harmonic Series corresponds to those steps of the tortoise in Zeno’s Paradox. So, not matter what kind of runner (even a runner with the speed of modern jet plane) held the race with the tortoise he will never catch up with it.
By the way, Robinson's non-standard analysis can do nothing to solve any of those suspended “infinite-infinitesimals paradoxes” either.
Sincerely yours,
Geng
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I am looking for a survey or an original article that discusses what is known about homogeneity in preordered (quasiordered) sets.
We know that in the case of posets (partially-ordered sets), the theory of automorphisms and homogeneity has reached a great extent.
Recall that, a poset P is called homogeneous, if for each two elements a, b of P there is at least one automorphism of P which maps a onto b.
You can look for example at chapter 10.4 of the book "Egbert Harzheim - Ordered Sets [Advances in Mathematics]".
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Can inconsistent or inother logic save naive set theory?
Dear Jaykov Foukzon.
I appreciate that there are many theories of have sprouted up as a result of the recognition the various paradoxes in set theory. To my understanding this is what initiated Hilbert's program and the "axiomatic set theory" initiative.
To clarify my statement, I referenced standard results, and not a standard model. It should be clear (under completeness), if there were a standard model, then there would be no inconsistencies and so the point of this question would be mute.
From you paper:
• ”But how do we know that ZF C is a consistent theory, free of contradictions? The short answer is that we don’t; it is a matter of faith (or of skepticism)”— E. Nelson wrote in his paper [1].
So I wonder if you can clarify for me, as I'm new to this, how you might suspect that ZFC is immune from paradoxes/contradictions if it has no model?
TIA
Jim
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There are three answers to the Set-Theory of Cantor.
Let´s have a discussion about them.
See first:
• Introduction Cantor
The papers, which the discussion should go on, are:
• Cantor
• Proof
• Cantor 3
The paper of Zenkin is a good basis for to discuss the difference to my papers.
Zenkin disproved the way Cantor went altogether, but failed in `using the binary number system for the real numbers presentation´. That only could be done by a bijective relation. In my opinion he didn't even do that. What do you mean?
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We went through an entire roundabout within this discussion only to realize that the initial proposed set of definitions are in use both in Computer Science and in Mathematics. For a simple initial set of citations kindly refer to https://en.wikipedia.org/wiki/Sequence (Sub-section Streams)
( Oflazer, Kemal. "FORMAL LANGUAGES, AUTOMATA AND COMPUTATION: DECIDABILITY" (PDF). cmu.edu. Carnegie-Mellon University. Retrieved 24 April 2015. ).
We had proposed an extension of this as we felt this to be a more intuitive when describing certain aspects of Cantor's Diagonal argument.
We were met with an influx of knee-jerk reactionary responses that one will find within this discussion claiming that such definitions do not exist or that it is 'gibberish' to make use of terms such as 'stream' . There is obvious evidence to the contrary, and one does not need to dig too deep in order to see this.
What we ask of the RG community is that any and all knee-jerk responses without understanding the intention of a question, not only makes the response seem unprofessional and un-researched bringing into question the integrity/knowledge of the responder, it has the added disadvantage of dissuading one from asking questions.
A feeling one gets is that such knee-jerk responses are a result of a need to display 'knowledge/superiority' over a field they feel they hold dominion over along with the assumption that the questioner's inability to phrase implies complete ignorance.
I feel a bit foolish, My apologies. I have revised.
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We make the following proposal to the Mathematical community.
Anyone well versed in Logic and Set Theory will be familiar with the concept of cardinality and the nature by which the cardinality of the real numbers R as a whole over the natural numbers N was established.
There is a clear reason why the diagonalisation argument [Diag(N,R)] works in the way of the above establishment.
Being able to pinpoint the reason, will have the effect of being able to prove \kappa{S_1} > \kappa{S_2} for sets S_1 and S_2 having the necessary characteristics, by performing : [Diag(S_1,S_2)].
To aide in clarifying the above, we have attached a diagram.
We will be presenting the above idea in a conference in India.
The idea of increasing sequences of equal types (zero´s) for creating excluded subsets is an idea of my papers (Cantor, Proof, Cantor 3).
My papers represent another conclusion but I will not allow you to present the main idea as your work.
The German versions of my papers were present at RG for the ask for help in translation (Who likes to tanslate the counter-proof of the diagonal slash argument from German to English? / Is contradiction between Cantor slash1 and slash2 already known?).
Please accept my copyright or I have to use all possible tools to hold my rights.
You may ask for an acceptance for quoting my papers.
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Dear Professors,
I can find many "AMS Classifications" for fuzzy set theory and related algebraic structures. And also for soft set theory "06D72" is available. Is any other classifications for fuzzy soft set theory?
Thank you...
Thank you sir @Ganesan G
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The well-known Zermelio's theorem states that every set can be well-ordered. Since arbitrary well-ordering is a linear ordering, from this theorem it follows the following corollary:
(A) An arbitrary set can be linearly ordered.
It is well-known that Zermelio's theorem is equivalent to the axiom of choice.
Question: Can Corollary (A) be proven without axiom of choice?
.
if you can make sense of the following thread :
(i'm having hard time with it ... but thanks for the question ; had me review a pile of long-forgotten concepts !)
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The European revolutions produced between 1789 and 1848 gave rise to a new type of state that historians call "liberal". The political philosophy that sustained these regimes is the so-called "liberalism", which in the mid-nineteenth century had a dual aspect: political and economic. Liberalism implies the respect to the citizen and individual liberties protected, in a general way, on an inviolable constitution that reflects the rights and duties of citizens and rulers; separation of legislative, executive and judicial powers to avoid any abuse of power, and the right to vote. Along with this political liberalism, the bourgeois state of the nineteenth century is also based on economic liberalism: a set of theories and practices at the service of the upper bourgeoisie and which, to a large extent, were a consequence of the industrial revolution. From the point of view of practice, economic liberalism meant the non-intervention of the state in social, financial and business issues. From here, and based on the experiences that we live in our countries, I propose this question. Thanks in advance for your responses.
It all depends on what is behind the concept of economic liberalism. If we look at what is happening in China then the market is compatible with the Maoist bureaucratic dictatorship. This is also true for Saudi Arabia. and for other countries. There is therefore no automatic and necessary link between the freedom of capital, goods, entrepreneurship and political freedoms. An important point is that of property rights. Without these rights there is no economic liberalism possible.
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（1）Denying the essential differences of infinite set elements’ “special nature, special existing condition, special manifestation and special relationship among each other”, so, elements in all different infinite sets are only heaps of infinite “indiscriminative, nonquantitative, abstract points” and it is unnecessary and impossibale to have any quantitative studying and cognizing on “infinite elements (points) in infinite sets”, the number of elements in all infinite sets are the same and it is just infinite. For example, elements in many infinite mother sets and their sub-sets are in fact all the same stuff of infinite “indiscriminative, nonquantitative, abstract points” without any differences of “nature, existing condition, manifestation and relationship among each other”. The typical cases are “the element numbers in Rational Numbers Set and Natural Numbers Set are equal, the element numbers in Natural Numbers Set and Even Numbers Set are equal and they are just all infinite”. The conclution is: any infinite set has limitless, endlees, infinite elements, so their “one-to-one coresponding operations” of quantitative studying and cognizing on “infinite elements (points) in infinite sets”
can be carried on for ever and their element numbers are all “equally infinite”.
（2）Affirming the essential differences of infinite set elements’ “special nature, special existing condition, special manifestation and special relationship among each other”, so, elements in all different infinite sets can be “discriminative, quantitative visible and tangible mathematical things” and it is necessary and possibale to have all kinds of quantitative studying and cognizing on “infinite elements in infinite sets”. For example, there are different “special nature, special existing condition, special manifestation and special relationship among each other” betwee elements in Real Numbers Set and Natural Numbers Set, so, these two infinite sets have different element numbers. The typical cases are “the element numbers in Real Numbers Set are more than that of Natural Numbers Set, the element numbers in Power Set are more than that of its original set and they are all inequal”. The conclution is: Different infinite sets may have different element numbers. So, in the “one-to-one coresponding operations” of quantitative studying and cognizing on infinite elements of two different infinite sets, the elements in smaller set with fewer elements are consumed and finished soon, it means the element numbers in such an infinite set are not endlees and limitless at all (fake infinite); but in the begger set, infinite many elements are left during this operations, this means its elements are endlees and limitless (real infinite), never be consumed and finished at all, the “one-to-one coresponding operations” here can never be carried on for ever at all.
（3）The above different operating ideas and results of quantitative studying and cognizing on “elements in infinite sets” in present classical infinite set theory are affirmed scientific and both aceptable. The one-to-one coresponding operation idea and result of quantitative studying and cognizing on Rational Numbers Set and Natural Numbers Set is the most typical example. In present classical infinite set theory, on the one hand, we can prove the general acepted conclution of “Rational Numbers Set has exactly same elements as Natural Numbers Set, so it is countable” by the above first operating idea and result; on the other hand, we can also prove “Rational Numbers Set has infinite more elements than Natural Numbers Set, so it is uncountable” by the above second operating idea and result, because just a tiny portion of rational numbers from Rational Number Set (such as 1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) can map and use up (bijective) all the numbers in Natural Number Set. This is the newly discoved “’Countable-uncountable’ Paradox of Rational Number Set”. The conclution is: the quantitative studying and cognizing theories and operations in present classical infinite set theory are lack of scientificity.
It is because of the fundermental defects of the absence of “infinite carrier theory” and the unscientific concepts of “potential infinite--actual infinite” in present classical infinite set theory that have been making people unable to know which of the above threedifferent operating ideas and results is scientific and why, unable to know scientifically and systematically the relationship between “elements in infinite set” and “infinite set” , unable to know scientifically and systematically the “nature, existing condition, manifestation and relationship among each other” of sets through their elements, ... This situation has been making people unable to know clearly since antiquity how to carry on quantitative studying and cognizing on elements in infinite set scientifically and systematically, resulting in the production and suspending of all kinds of “infinite things’ quantitative cognizing paradox families” in present classical infinite set theory.
Five of my published papers have been up loaded onto RG to answer such questions:
1，On the Quantitative Cognitions to “Infinite Things” (I)
2，On the Quantitative Cognitions to “Infinite Things” (II)
3，On the Quantitative Cognitions to “Infinite Things” (III)
4 On the Quantitative Cognitions to “Infinite Things” (IV)
5 On the Quantitative Cognitions to “Infinite Things” (V)
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The suspended “infinite related paradoxes” in analysis and the “infinite related paradoxes” in set theory used to be studied and discussed separately since ancient times, it used to be regarded that they have been nothing to do with each other.
But, if putting all the suspended “infinite related paradox families” in analysis and set theory together, we discover immediately that they are in fact forming an “‘infinite paradox’ symptom complex” in present infinite related areas of mathematics disclosing exactly the same fundamental defects in present infinite related science and mathematics.
Then, we discover a new way to study and solve the fundamental defects in present infinite related science and mathematics disclosed by this “‘infinite paradox’ symptom complex” -------a new working field is waiting to be opened up.
Five of my published papers have been up loaded onto RG to answer such questions:
1，On the Quantitative Cognitions to “Infinite Things” (I)
2，On the Quantitative Cognitions to “Infinite Things” (II)
3，On the Quantitative Cognitions to “Infinite Things” (III)
4 On the Quantitative Cognitions to “Infinite Things” (IV)
5 On the Quantitative Cognitions to “Infinite Things” (V)
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I was trying to code the basic procedure for attribute reduction using rough set theory... couldn't find any pseudo code to find reduct and core...
I suggest take a look a this
really simple explanation
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Dear colleagues
I am trying to acquire the graph of probability of failure versus factor of safety. In other words, I would like to depict a figure in which the vertical axis shows the probability of failure and the horizontal axis shows factor of safety.  As a result of random set theory, the cumulative density function for factor of safety has been calculated. That is to say, there has been a graph in which the vertical axes shows cumulative density function and the horizontal axis shows factor of safety. The question arises here, how the probability of failure for each factor of safety would be derived from cumulative density graph of factor of safety?
Should I use fragility function in order to obtain probability of failure?
I would be pleased to give you any further information that you may request.
By definition, a slope fails with a FoS going below 1.0. As long as the FoS is above 1.0, a slope is considered stable. Thus, I do not see how your figure could be drawn.
Of course, the FoS is a simplification and can not be measured directly. However, the definition is rather clear and if you follow it a FoS>1 relates to zero probability of failure.
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For almost all the set models defined in literature, order relations have been defined so that elements can be compared. Do we have any such order relations defined so that we can have neutrosophic lattice or neutrosophic Boolean algebra etc. This will lead to many ordered algebraic structures and neutrosophic Boolean algebra will lead to neutrosophic circuits and so on.
I think there is not an ordered relation that is defined on neutrosophic sets.
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For the sake of interest I was out looking for a discussion of functions in computer science as generalized functions. I started my search by something such as "computer functions as mathematical functions". NO paper discusses functions in computer science in terms of set theory, and they compare (not equate) mathematical functions and functions in programming. Where did I miss the buss? Mathematics need not be done with numbers, although the largest application of mathematics is to numbers.
Short answer. The use of the term "function" as a designator in commonly-used programming languages is actually a mistaken use of the mathematical notion having the same name. It seems that, in the early days of computer software and programming language development, the choice of nomenclature was in some sense a corruption of the mathematical usage. It wasn't malicious, but an over-reaching that was perhaps not well-understood at the time.
In programming languages, "function" is used to designate a specific computational procedure. For appropriately-written procedures, there is establishment of a specific algorithm for some similarly-named mathematical function. It is not the function, it is an algorithmic procedure. When accurate, the correspondence between represented operands and the represented result satisfies the relationship established for the mathematical functions domain and range members.
There are many procedures for the same function, in this sense (and there can be many mathematical characterizations of the same function).
The clouding of nomenclature becomes an issue when one needs to deal with the fact that mathematical use of functions need not have any direct connection with computation.
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Fuzzy set theory; Rough set theory; Applicability of set theory; Machine learning
Rough and Fuzzy set are almost same application wise. But theoretically they are different which makes Rough Sets superior than Fuzzy (Personal Opinion). Let me explain why theoretically Rough is better than Fuzzy. Fuzzy set starts with identifying a membership function a-priori and tries to fit the data in its theory, whereas Rough Set starts with no such assumption on membership function. Rough sets straightway starts fitting the data blindly from which membership function values are computed. This is why Rough sets make better explanation of uncertainty as it mimics what the data speak. As per my opinion Rough set is better suited in case of data science where prior information and knowledge about the process under consideration are not available and the analysts have to rely purely on the data.
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The studies to our infinite related science and mathematics history have proved that a new visible and tangible “infinite related mathematical carrier (called ‘infinite mathematical carrier’ or ‘infinite carrier’ for short following)” is needed in our science and mathematics to express and cognize quantitatively “infinite related mathematical things”. This special new “infinite carrier” is a new mathematical content (new mathematical manifestation, new mathematical entity) between “infinite things” and “finite things”, its emerging and existence decides the emerging and existence of “infinite related mathematical carrier theory”-------new manifestations of “mathematical carrier” (such as “inter-number，indistinct-number” and “inter-set，indistinct-set”) and their related quantitative cognizing and operating theory (such as limit theory and one-to-one coresponding theory) as well as new infinite related “number spectrum (number system)” and “set spectrum (set system)”. The developing of “infinite related mathematical carrier theory” leads us enter into a new and systematic cognizing field of infinite carrier, enables us to accomplish the scientific mission of “cognizing ‘abstract law’ with ‘concrete carriers’” efficently, systematically and scientifically (fully satisfied accurace with high deree of approximation) to all kinds of “infinite related mathematical things” in many quantitative cognizing situations.
The emerging and existence of new mathematical carrier “inter-number (indistinct-number)” has decided the generating of the new infinite related “number spectrum--------zero or absolutely none, infinitesimal, inter-small or indistinct-small, finite-number or distinct-number, inter-great or indistinct-great, infinity, yan or absolutely being" for solving the fatal fundamental defect of lacking “infinite mathematical carrier” in the process of quantitative cognizing to “infinite related number forms”; while the emerging and existence of new mathematical carrier “inter-set，indistinct-set” has decided the generating of the new infinite related “set spectrum---------empty set，finite-set, distinct-set，inter-set or indistinct-set，infinite set" for solving some other fatal fundamental defect of lacking “infinite mathematical carrier” in the process of quantitative cognizings to “different infinite things relating to set” since set theory was founded. Our studies proved that the abscence of the whole “infinite related mathematical carrier theory” is really one of the key factors for the emerging and existence of those suspended paradox families (‘infinite paradoxes’ symptom complex) in present set theory and analysis.
In previous discussion we have introduced the newly constructing “new infinite theory system as well as its related infinite mathematical carrier theory” basing on the accumulated quantitative cognizing fruits of thousands years work on “infinite related mathematical things” by our predecessors. As one of the infinite related mathematical carriers, the “inter-set” in “set spectrum” of new infinite set theory are new mathematical contents of visible and tangible “scientific carriers” (in fact, many infinite sets being quantitative cognized in present set theory are “inter-sets” which have very close relationship with their elements’ special “nature, existing condition, manifestation and relationship among each other”). And, with the related “quantitative treating theories of infinite related mathematical things” (such as those “carriers’ analysing and quantities reaching” theories to elements and number forms, limit theory, one-to-one coresponding theory, ...), we can cognize on “inter-sets” and attain our objective of scientific, systimatic cognitions to all kinds of sets in the infinite related new “set spectrum”.
it is because of the absence of the infinite related new number forms and set forms (infinite related things mathematical carriers) that makes us unable to treat “different beings of infinite related things” in mathematical analysis and set theory ever since.
Infinite refers to "infinite" or "infinite" or "unlimited" used in several different concepts but all of which combine one idea of ​​"no end". In this sense, it is related to philosophy, mathematics, divinity, and life Daily also.
The first known symbol ()) for this expression was John Wallis in 1655 in his writings: De Sectionibus Conicis and Arithmetica Infinitorum.
In popular culture, infinity is usually something that can be likened to "as many as possible" or as far as possible. In many minds, the question remains: what is after infinity, but much is considered a question after infinity is absurd because infinity is a symbol of what can not be imagined Is greater than it.
In mathematics, infinity is used as a number to be measured by an unlimited quantity, symbolized by a letter ()). It is an entity different from any other numerical entity in its characteristics and behavior.
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How can we define “infinite” and “finite” if we agree with the idea and the operations in Cantor's proofs that many infinite sets in mathematics actually can be proved (turned into) finite set-------the elements in real number set are never-be-finished, endless, limitless and they are really infinite while those in natural number set are sure-be-finished, ended, limited and they are actually finite?
This conundrum has disclosed two infinite related fundamental defects in present infinite set theory and mathematical analysis:
（1）the theoretical and practical confusions of “finite--infinite” caused by the theoratical and practical confusions of “potential infinite, actual infinite”;
（2）the absence of systematic “scientific carrier theory” and the absence of systematic “infinite related number spectrum and set spectrum”.
Because of the absence of systematic “scientific carrier theory”, in present infinite theory system (The First Generation of Infinite Theory System) basing on “potential infinite, actual infinite”, people have had to use the unscientific “potential infinite, actual infinite” theory for the quantitative cognitions of those “finite--infinite” related mathematical things since antiquity. And, because all the “infinite related concepts and contents” for quantitative cognizing in our mathematics have been being unscientifically defined as “non—infinite infinite, finite infinite, infinite finite, both finite and infinite, number of non-number, ...” in present infinite theory system, no one is able to know clearly when and where they should be “finite or infinite, number or non-number, ...” and why. This situation is not only unable us to cognize scientifically two concepts of “finite, infinite” and the relationship between them, but also unable us to cognize scientifically three contents of “finite mathematical things, potential infinite mathematical things, actual infinite mathematical things” and the relationship among them in present infinite related mathematical fields------very free and arbitrary when dealing with any “infinite related mathematical things”. So, many suspended infinite related paradox families have been produced inevitably. It is because of such fatal fundamental defects that results in the frequently performings of the mathematical version of Andersen's THE EMPEROR'S NEW CLOTHES Fairy Tale in all infinite related mathematical fields ever since “potential infinite, actual infinite” came into our science and mathematics, the only thing most people are able to do is just echoing in earnest to what the authorities said.
In new infinite theory system basing on the new concepts of "abstract infinite law and carriers of abstract infinite law " (The Second Generation of Infinite Theory System), the concepts of “finite, infinite, finite scientific carrier, infinite scientific carrier” have been clearlly and scientifically defined. “Infinite” is a kind of “invisible and intangible abstract scientific law”, it is a scientific (mathematical) content for qualitative but not quantitative studies and cognizings; while the “abstract scientific infinite law carriers" is a kind of “visible and tangible scientific concrete manifestation (entity) of abstract scientific law”, it is a scientific (mathematical) content specially fot quantitative studies and cognizings. In our science, we can only conduct the quantitative studying and cognizing on the “abstract scientific law carriers" but not on the “abstract scientific law". Basing on new infinite theory system, now we are perfectly justifiable to construct a systematic and scientific “theory of infinite related mathematical carriers" and able to conduct quantitative studying and cognizing on all kinds of “infinite related mathematical carriers" scientifically (such as their computations as well as the comparetions among them). The quantitative related systematic and scientific “theory of carriers" decides that we must have quantitative studies and cognizings on “different infinite law carriers, big (small) infinite law carriers, bigger (smaller) infinite law carriers, more bigger (more smaller) infinite law carriers, ..." but not on “different infinite laws, big (small) infinite laws, bigger (smaller) infinite laws, more bigger (more smaller) infinite laws, ..." or even arbitrarily on the “finite—infinite mixing up things” as what have been done in presentmathematical analysis and set theory (such as taking them as finite things first then as infinite things later or taking them as infinite things first then as finite things later or the elements belonging to an infinite set but being impossible to exist in this very infinite set, ...). So, in the Second Generation of Set Theory, it is impossible at all to have such contents as “the infinite of more infinite, more more infinite, more more more infinite, ... super infinite, super super infinite, super super super infinite, super super super super infinite, ...”; All the mathematical contents of “X--->0, Y--->📷, ...” for quantitative cognizing in our new infinite related mathematical analysis and set theory are the “scientific carriers" with clear and scientific quantitative natures and definitions in new systematic and scientific “infinite related number spectrum (number system) and set spectrum (set system)”. The replacement of “potential infinite, actual infinite” concepts by new "abstract infinite law and carriers of abstract infinite law" concepts in present infinite theory system and the development of systematic “theory of scientific carriers” inevitably result in great theoretical and operational reformations (a revolution) in the quantitative--quantitative studies and cognizings in our infinite related science areas and, eliminate scientifically and thoroughly the suspended conundrums of “’potential infinite, actual infinite’ confusing" and “’finit--infinite’ confusing" since antiquity.
Gang, I think that problem is ( partially) with the use of a language. To get clear understanding what a particular language expression mean you need a higher ( the second) level f language ,or meta language to describe the first level language . This in turn will call for a meta meta language and so on. Those are difficult problems and I have not enough expertise in that field to comment any further.
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The unexploited unification of general relativity and quantum physics is a painstaking issue. Is it feasible to build a nonempty set, with a binary operation defined on it, that encompasses both the theories as subsets, making it possible to join together two of their most dissimilar marks, i.e., the commutativity detectable in our macroscopic relativistic world and the non-commutativity detectable in the quantum, microscopic world? Could the gravitational field be the physical counterpart able to throw a bridge between relativity and quantum mechanics? It is feasible that gravity stands for an operator able to reduce the countless orthonormal bases required by quantum mechanics to just one, i.e., the relativistic basis of an observer located in a single cosmic area?
What do you think?
Continuing the discussion about non-commutatity and spacetime, see
and
More to the point, there is the issue of inflation in non-commutative space time, introduced in
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Paradoxes in our science have been special scientific contents which are constructed purpersly or unpurpersly to disclose the fundamental defects in certain areas of humans science. Along with humans’ evolution and the expension of our knowledge, our science has been evluting and completing constantly. So, paradoxes have been accompanying our science all the time-------no fundamental defects no paradoxes.
Our science history has proved that the existence of paradoxes in certain area of human science is not only the touchstone for the defects in the very science area (such as the suspended paradoxes in present infinite set theory) but also the strong “metabolism signal” for the related basic theory. And, the “‘infinite related paradoxes’ symptom complex” in some science areas (such as the suspended paradox families in present mathematical analysis and infinite set theory) disclose even more in their special way the ponderance and universality of the fundamental defects in their related science areas -------new scientific basic theories must be constructed to replace those former “paradoxes generating unscientific basic theories” in those very science areas.
Those suspended “infinite related paradoxes” in present mathematical analysis and in set theory used to be studied and discussed separately since ancient times, it used to be regarded that they have been nothing to do with each other. But, if putting all these paradoxes in analysis and in set theory together, we discover immediately that they are in fact forming an “‘infinite related paradoxes’ symptom complex” discloses in its special way exactly the same fundamental defects in different areas of present infinite related science and mathematics. This new idea helps us open up a new way to study and solve those suspended “infinite related paradoxes” in present mathematical analysis and in set theory-------discarding the inevitably confusing concepts of “potential infinite--actual infinite”, developing the new infinite theory system basing on the theory of "abstract infinite law--abstract infinite law carriers" and looking for the scientific foundation for mathematical analysis and in set theory.
Thank you very much for your ideas dear Mr. Peter Kepp，
Studies on the infinite related mathematics history have proved that it is the inevitable fundamental defect of “’potential infinite--actual infinite’ confusing” caused by the absence of systematic theory of “scientific carriers” that decides any areas in present infinite related science (including mathematical analysis and set theory) are unable to escape from the influences and constraints of such fatal defect, unable us to conduct the quantitative cognizings scientifically to “infinite related mathematical things”. So, many infinite related paradox families (symptom complex) in present mathematical analysis and set theory have been inevitably produced. Although many ways to solve this “paradoxes symptom complex” have been invented in our infinite related mathematics (such as Standard Analysis, Nonstandard Analysis, Type Theory, Model Theory, ZFC ) since Zeno 's Achilles and Tortoise Paradox, non of them is able to solve the real trouble of “’potential infinite-actual infinite’ confusing”--------The Second Mathematical Crisis and The Third Mathematical Crisis are unsolvable at all in present infinite related science and mathematics basing on classical infinite theory system. The systematic studies and development of “the idea and theory of ‘scientific abstract law--the carriers of scientific abstract law’” (especially the systematic studies and development of “the idea and theory of ‘the carriers of science’”) will fill in a blank in the foundation of Logic Theory and open a new way and a new working field for our further studying and cognizing on Logic (the relation laws among things in the universe).
“The natural metabolism law (mechanism)” is governing the whole life process of any “alive things” in the world. Humans’ mathematics is an alive thing concomitant humans, it metabolized and evolved before, it is metabolizing and evolving now, it will metabolize and evolve in the future and, those unscientific things must be out while some scientific things must be in whenever the time is ripe. Having been suffering from the tormenting of “‘potential infinite--actual infinite’ confusing”, we should look for the new working idea and jump out of the “‘potential infinite--actual infinite’ confusing” abyss. The emergence of the new infinite theory system and the revolution in the foundation of present mathematical analysis and set theory decide the emergence of the Fourth Generation of Mathematical Analysis and the Second Generation of Set Theory. Only discarding the unscientific concepts of “potential infinite, actual infinite” in present infinite theory system and construting the new infinite theory system basing on the scientific concepts of "abstract infinite law--abstract infinite law carriers", studing and developing the systematic new "’scientific carriers--scientific carriers treating’ theories", can we fairnessly and justifiably conducting the quantitative cognitions to “infinite related mathematical things” with the systematic theory of “scientific carriers”, solving the fundamental defects disclosed by the obstinate “infinite paradoxes symptom complex” theoretically and operationally to dispel the thousands years suspended black cloud of “infinite related paradox families” over the sky of present mathematics and science.
Yous,
Geng Ouyang
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Studies on the infinite related mathematics history have proved that it is the inevitable fundamental defect of “’potential infinite--actual infinite’ confusing” caused by the absence of systematic theory of “scientific carriers” that decides any areas in present infinite related science (including mathematical analysis and set theory) are unable to escape from the influences and constraints of such fatal defect, unable us to conduct the quantitative cognizings scientifically to “infinite related mathematical things”. So, many infinite related paradox families (symptom complex) in present mathematical analysis and set theory have been inevitably produced. Although many ways to solve this “paradoxes symptom complex” have been invented in our infinite related mathematics (such as Standard Analysis, Nonstandard Analysis, Type Theory, Model Theory, ZFC ) since Zeno 's Achilles and Tortoise Paradox, non of them is able to solve the real trouble of “’potential infinite-actual infinite’ confusing”--------The Second Mathematical Crisis and The Third Mathematical Crisis are unsolvable at all in present infinite related science and mathematics basing on classical infinite theory system. The systematic studies and development of “the idea and theory of ‘scientific abstract law--the carriers of scientific abstract law’” (especially the systematic studies and development of “the idea and theory of ‘the carriers of science’”) will fill in a blank in the foundation of Logic Theory and open a new way and a new working field for our further studying and cognizing on Logic (the relation laws among things in the universe).
“The natural metabolism law (mechanism)” is governing the whole life process of any “alive things” in the world. Humans’ mathematics is an alive thing, it metabolized before, it is metabolizing now, it will metabolize in the future and those unscientific things must be out while some scientific things must be in whenever the time is ripe. Having been suffering from the tormenting of “‘potential infinite--actual infinite’ confusing”, we should look for the new working idea and jump out of the “‘potential infinite--actual infinite’ confusing” abyss. The emergence of the new infinite theory system and the revolution in the foundation of present mathematical analysis and set theory decide the emergence of the Fourth Generation of Mathematical Analysis and the Second Generation of Set Theory. Only discarding the unscientific concepts of “potential infinite, actual infinite” in present infinite theory system and construting the new infinite theory system basing on the scientific concepts of "abstract infinite law--abstract infinite law carriers", studing and developing the systematic new "’scientific carriers--scientific carriers treating’ theories", can we fairnessly and justifiably conducting the quantitative cognitions to “infinite related mathematical things” with the systematic theory of “scientific carriers”, solving the fundamental defects disclosed by the obstinate “infinite paradoxes symptom complex” theoretically and operationally to dispel the thousands years suspended black cloud of “infinite related paradox families” over the sky of present mathematics and science.
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space) . Also, set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. The theory had the revolutionary aspect of treating infinite sets as mathematical objects that are on an equal footing with those that can be constructed in a finite number of steps. Since antiquity, a majority of mathematicians had carefully avoided the introduction into their arguments of the actual infinite (i.e., of sets containing an infinity of objects conceived as existing simultaneously, at least in thought). Since this attitude persisted until almost the end of the 19th century, Cantor’s work was the subject of much criticism to the effect that it dealt with fictions—indeed, that it encroached on the domain of philosophersand violated the principles of religion. Once applications to analysis began to be found, however, attitudes began to change, and by the 1890s Cantor’s ideas and results were gaining acceptance. By 1900, set theory was recognized as a distinct branch of mathematics.
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I wrote a note (available as preprint on Research Gate) in which I reconsider Cantor’s diagonal argument for the existence of uncountable sets from a different point of view. After reformulating well-known theoretical results in new terms, I show that, contrary to what stated by Cantor, they do not imply uncountability.
To be clear, the aim of the note is not to prove that R is countable, but that the proof technique does not work. I remind that about 20 years before this proof based on diagonal argument, Cantor gave another and totally different proof of the same result.
I woud like to have comments from experts.
This is a very good question. In addition to what has already been observed, there is a bit more to add.
First, here is a followup to @ Peter Kepp 's comment. His paper is available at
See. also:
Three very interesting, related papers by @Wolfgang Mueckenheim are
and
Also of interest is the paper by A. Zenkin:
Perhaps followers of this thread will also be interested in Wittgenstein's diagonal argument:
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The fundamental defects of “potential infinite and actual infinite” confusions in present classical infinite set theory have been making us humans unable to study and cognize scientifically the foundation of “one-to-one correspondence theory” (the “one-to-one correspondence theory needs its own foundation” even have never been considered about). And, because of the absence of this very foundation, it is very difficult for people to really understand scientifically what kind of mathematical tool “one-to-one correspondence theory” is and how to operate with this mathematical tool in practical quantitative cognitions to elements in infinite sets. So, following five questions have been produced and troubling people long:
（1）Are the elements in infinite sets “potential infinite things” or “actual infinite things”？
（2）Are there different “one-to-one correspondence theories and operations” to “potential infinite elements” or “actual infinite elements” in infinite set theory？
（3）How do we practically carry on “one-to-one correspondence” operations between two sets-------do we have “‘one single element’ to ‘one single element’ correspondence” or “‘one single element’ to ‘many elements’ correspondence” or “‘many elements’ to ‘many elements’ correspondence”？can we arbitrarily alter the elements’ “special nature, special existing condition, special manifestation and special relationship among each other” in infinite sets during the “one-to-one correspondence operations” for quantitative cognitions (such as alter all the elements in Natural Number Set first [1x2, 2x2, 3x2, 4x2, …,nx2, …] = [2,4,6,8, …,e,…] (not the correspondence between N and E but E andＥ), then prove it has same quantity of elements in Even Number Set)?
（4）What kinds of the elements in two different infinite sets are corresponded-------- do we have “‘one single original element’ to ‘one single original element’ correspondence” or “‘actual infinite elements’ to ‘potential infinite elements’ correspondence” or “mixture correspondence of ‘actual infinite elements’ and ‘potential infinite elements’”？
（5） What on earth is the foundation of “one-to-one correspondence theory”？
The fundamental defects in present classical infinite set theory have made us unable at all to answer clearly and scientifically above five questions. So, when carrying on practical quantitative cognitions to elements in different infinite sets with “one-to-one correspondence theory”, one can do very freely and arbitrarily--------lacking of scientific basis. For example: it is because of acknowledging the differences of elements’ “special nature, special existing condition, special manifestation and special relationship among each other” between Real Number Set (R) and Natural Number Set (N), one can prove that the Real Number Set (R) has more elements than N (the Power Set Theorem is proved in the same way). But, as what has been discussed in above 2.1 .1, we are able to prove with exactly the same way “the mother set has more elements than its sub-set”, “Rational Number Set has more elements than Natural Number Set”, “Natural Number Set has more elements than odd number set” ,...; we can even apply the widely acknowledged method of altering elements’ “special nature, special existing condition, special manifestation and special relationship among each other” to prove “Natural Number Set has more elements than Natural Number Set”, “odd number set has more elements than even number set”, “even number set has more elements than odd number set”, ....
Basing on the new infinite theory system with the “infinite mathematical carriers theory”, the Second Generation of Set Theory provides us with the scientific foundation of “one-to-one correspondence theory” and enable us answer above five questions clearly and scientifically:
（1）the elements in infinite sets are “infinite related mathematical carriers” with explicit quantitative nature and definition, indicating the existing of “abstract infinite law” and nothing to do at all with “potential infinite--actual infinite”. This decides one of the major differences between the first and the second generation of set theories-------the elements in different infinite sets have their own “special nature, special existing condition, special manifestation and special relationship among each other. So, it is really possible that different infinite sets have different quantity of elements and people can take them really as “visible and tangible infinite related mathematical things (such as the new numbers in new number spectrum)” for the quantitative cognitions
（2）the elements in infinite sets have nothing to do at all with “potential infinite elements” and “actual infinite elements”, there is only one identity for them-------“infinite related mathematical carriers” with explicit quantitative nature and definition; So, there is only one “one-to-one correspondence theory and operation” for them.
（3）it is explicitly stipulated that only “‘one single original element’ to ‘one single original element’ correspondence” operation is scientific (allowed) when comparing two sets for the quantitative cognitions and, during this process, any operations of arbitrarily altering the elements’ “special nature, special existing condition, special manifestation and special relationship among each other” are unscientific (not allowed).
（4）in the Second Generation of Set Theory, because of nothing to do at all with “potential infinite--actual infinite”, it is impossible to have any troubles produced by the confusion of “potential infinite --actual infinite”.
（5）the new infinite theory system (especially its theory of “infinite related mathematical carriers”) is the foundation of “one-to-one correspondence theory”.
Pls see bour book Intermediate Set Theory, F R Drake and D Singh, Chapter 1,etc. John Wiley, 1996.
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The emergence of new infinite system determines the production of "new infinite set theory"--------The infinite set theory based on the classical infinite system is called "classical infinite set theory (the First Generation Of Infinite Set Theory) " while the infinite set theory based on the new infinite system is called "new infinite set theory (the Second Generation Of Infinite Set Theory)".
The same mission and same cognizing contents decide the two similarities between the two set theories.
1 The same qualitative-quantitative cognizing motivation and idea
It is a must to carry on the qualitative-quantitative cognizing activities on “infinite related mathematical things (such as elements in infinite sets and the quantity of elements in infinite sets)” by both new and classical set theories; especially in many practical quantitative cognizing operations, most “mathematical contents” in infinite sets are treated as “mathematical things with visibal and tangible quantitative nature and meaning” by both new and classical set theories.
2 the same quantitative cognizing tools
Both new and classical set theories use one-to-one coresponding theory and limit theory to carry on quantitative cognitions to those “infinite related mathematical things” in infinite sets.
It is these two similarities between new and classical set theories that decides many invaluableners intellectual wealth accumulated since antiquity in classical set theory are reserved in new set theory.
Dear Geng,
I can't quite understand your language. The survival of Mathematical theories does not depend on their ages, but their consistency. This is why I cannot consider the terms "old" and "new" as logical axioms of validity.
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We understand from our infinite related science history that in present classical infinite related science system, it has been admitted that the concept of infinite is composed by both “potential infinite” and “actual infinite”. On the one hand, no one is able to deny the qualitative differences and the important roles “potential infinite--actual infinite” play in the foundation of present classical infinite theory system; on the other hand, no one is able to deny that the present classical set theory is basing on “potential infinite--actual infinite” concepts as well as its related whole present classical infinite theory system. The fact is: any areas in present classical infinite related science system (of cause including present classical mathematical analysis and set theory) can not run away from the constraining of “potential infinite--actual infinite” concepts-------all the contents in present classical mathematical analysis and set theory can only be existing in the forms of “potential infinite mathematical things” and “actual infinite mathematical things”. But, the studies of our infinite related science history have proved that no clear definitions for these two concepts of “potential infinite--actual infinite” and their relating “potential infinite mathematical things--actual infinite mathematical things” have ever been given since antiquity, thus naturally lead to following two unavoidable fatal defects in present classical set theory:
（1）theoretically: It is impossible to understand what the important basic concepts of “potential infinite” and “actual infinite” and their relating “potential infinite number forms, potential infinite sets” and “actual infinite number forms, actual infinite sets” are and what kinds of relationship among them are. So, in many “qualitative cognizing activities on infinite relating mathematical things (such as all kinds of infinite sets, elements in infinite sets, numbers of elements in infinite sets)” in present classical set theory, many people even don’t know or actually deny the being of “potential infinite” and “actual infinite” concepts as well as their relating “potential infinite number form, potential infinite sets” and “actual infinite number forms, actual infinite sets”--------it is impossible at all to understand clearly and scientifically the exact relationship among the important basic concepts of “infinite, infinities, infinite many, infinitesimals, infinite sets, elements in infinite sets, numbers of elements in infinite sets”, ... So, it is impossible at all to understand clearly and scientifically all kinds of different infinite sets (such as lacking of the “’set spectrum’ for the overall qualitative cognitions on the existing forms of infinite sets”), elements in an infinite set (such as ”are the infinite related elements potential infinite mathematical things or actual infinite mathematical things, how they exist?”), numbers of elements in an infinite set (such as ”are they actual infinite many or potential infinite many?”), the “one-to-one corresponding theory and operation” in infinite sets (such as ”are the potential infinite elements corresponding to potential infinite elements or actual infinite elements corresponding to actual infinite elements or actual infinite elements corresponding to potential infinite elements?”) ,... --------the unavoidable defects of qualitative cognition on infinite sets and their elements
（2）operationally: First, it is impossible to understand whether the “elements in an infinite set, numbers of elements in an infinite set and all kinds of infinite sets” being cognized in present classical set theory are “potential infinite mathematical things” or “actual infinite mathematical things”, whether there are different theories and operations for “potential infinite mathematical things or actual infinite mathematical things”, and it is impossible at all to understand correctly (scientifically) in present classical set theory the natures of infinite related quantitative cognizing theories and tools (such as limit theory and the “one-to-one corresponding theory”) and their operational scientificities-------- it is impossible at all to master correctly (scientifically) the operational competences and skills of limit theory and the “one-to-one corresponding theory” thus resulting in no scientific guarantee for the operations of limit theory and the “one-to-one corresponding theory”; second, it is impossible at all to judge the scientificities of many infinite related quantitative cognizing activities in present classical set theory, people in many cases can only parrot every bit of what have been done by others or do as one wishes to treat many “not—knowing—what” infinite mathematical things with the unified way of “flow line” (any “infinite sets”, “elements of an infinite set”, “elements’ number of an infinite set” can either be “potential infinite” or “actual infinite”, neither be “potential infinite” nor “actual infinite”, first “potential infinite” then “actual infinite”, first “actual infinite” then “potential infinite”, ,,,), those believed and accepted Russell’s Paradox, Hilbert Hotel Paradox, Cantor’s operations of “cutting an infinite thing into pieces to make super infinite numbers with different grades” as well as the famous “using Russell's ‘paradox constructing mechanism’ to prove the Power Set Theorem and the uncountability of real number set” are typical examples of “potential infinite--actual infinite” confusing operations--------the unavoidable defects of quantitative cognition on infinite sets and their elements.
yes verey interesting... At first glance to understand infinity actual or potential are not easy. For example on the realline betweeen two so so close numbers there are infinitely many numbers....
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We understand from our science and mathematics history, the fundamental defect in present classical set theory disclosed by the members of Russell’s Paradox Family in different periods of time is: looking for something belongs to an infinite set but is impossible to be found inside this infinite set--------no logic in our science can solve such paradox family as all the members of Russell’s Paradox Family are produced by the confusion of “potential infinite” and “actual infinite”.
Even more, some members of Russell’s Paradox Family are taken as paradoxes while some are taken as “great theorems and axioms” .
Some people really believe that Russell's theory of types or ZFC really can solve Russell’s Paradox, but following case (two members of Russell’s Paradox Family) tells that neither Russell's theory of types nor ZFC works:
Applying exactly the same mechnisim of Russell’s Paradox------looking for something belongs to an infinite set but is impossible to be found inside this infinite set, Cantor proved two inportant results of “power set theorem” and “Real number set is uncontable”.
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I think Russell's Paradox Family proves that just because a mathematical concept is logically consistent, that does not mean it can be translated successfully into the real world. The actual infinite is logically consistent, but good philosophical arguments exist against the real existence of an actual infinite. In short, logical possibility does not guarantee metaphysical possibility.
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I'm currently teaching our intro course on Theoretical Computer Science to first-year students, and I came up with the following simple proof that there are non-computable functions from N->N:
It's a simple cardinality-based argument. The set of programs in any arbitrary language is countable - just enumerate by length-lexicographic order (shortest programs first, ASCIIbetically for programs of equal length). The set of functions from N->N is uncountable (as the powerset of N is uncountable - in a pinch by a diagonalization argument - and hence the set of indicator functions alone is uncountable). So there are more functions than programs. Hence there are functions not computed by any program, hence non-computable functions.
My question: This is so simple that either it should be well-known or that it has a fatal flaw I have overlooked. In the first case, can someone point me to a suitable reference? In the second: What is the flaw?
Thanks. Yes, I use the Church/Turing hypothesis/definition of computable (with any Turing-complete language, though programs for UTM are most traditional). OK, I see why I need the Axiom of Choice. I don't think I need the powerset axiom - that's just a convenient shortcut. And yes, it's not a constructive proof.
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Let suppose we say that there is a document D and a List of Sensitive Terms S. all NPs are belongs to D. Now IF are those NPs that belongs to S.
(1) IF = All NPs that belongs S.
Now Let suppose we have list A and B. Where A contains Identifiers and B contain medical finding.
All those NPs that is in A are I and All those NPs that is in B are F.
(2) I = { i E D | i E {a E A }}
(3) F = {f E D | f E {b E B}}
So implies that
(4) IF = I = { i E D | i E {a E A }} | F = {f E D | f E {b E B}}
or
(5) IF = I + F