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# Mathematical Logic - Science topic

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Is paraphrasing necessary in such cases, or is direct quotation with appropriate citation sufficient?
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Actually neither. One pr9v8des the reference to the work, where the equation appezred and explains how it’s being used in the new work.
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How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?
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Regards,
Germán Benitez Monsalve
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The idea is introducing student to the topic from a more general subject, and introduce the different structures of, say, Propositional Logic as a formal system, and from them deduce the Boolean Algebra (while introducing the interpretation of formulas), and further logical laws.
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I believe it is. I have taught logic myself to computer science students for several years.
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Prove or disprove that for every field F of characteristic two, there exist vector spaces U and V over F and mapping from U to V, which is F-homogeneous but not additive.
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Will this be the final incarnation of this question?
My purpose in asking these questions is to motivate the kind of physical theory that accepts that Physical Laws are part of The Universe, as opposed to standing outside it. And that rules governing The Universe must stem form The Universe itself. Otherwise, we should be asking: Where do the Physical Laws come from?
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"Laws of the Universe" is ambiguous between regulative properties and statements about regulative properties. If laws are aspects, tendencies, or dispositional properties of the universe then they are just that, namely physical features. Self-reference and logical circularity can only arise in referential symbol systems that are used to describe or model the universe and its dispositional properties.
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Is a physical basis that necessarily requires constancy of the speed of light a logical impossibility, or is the constancy of the speed of light the result of ideas not yet found or applied?
Does isotropy require constancy of the speed of light?
Jensen’s inequality for concave and convex functions, implies for a logarithmic function maximal value when the base of the log is the system’s mean. Mathematically, this implies that the speed of light must be uniform in all directions to optimize distribution of energy. This idea has a flaw. Creation of the universe happened considerably before mathematics and before Jensen’s inequality in 1906. Invert the conceptual reference frame and suppose that Jensen’s inequality is mathematically provable in our universe because it is exactly the type of universe that makes Jensen’s inequality mathematically true in it. A mathematical argument based on Jensen’s inequality goes around in a circle. Are there reasons, leaving aside Jensen’s inequality (or even including Jensen’s inequality), that require constancy of the speed of light?
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The discussion is about the light speed in the same medium(in general, the space vacuum). For example, does the light speed affected by the gravity
(Diffraction phenomenon) when passing near black holes? It has no meaning to study the light speed in a medium where light can not penetrate.
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I need to look at the value of the string variable for each entity to determine which distribution to use to assign my next variable. I cannot figure out how to do this, I either get the error that Symbol name "if" is a reserved name, or I get "invalid expression the value entered must be in the form of a mathematical or logical expression. String values are entered by enclosing a sequence of characters in quotation marks."
Maybe I am using the wrong format, or I am not allowed to write such complex statements in the new value field of the assign block. I currently have:
if(Scale == "200pm")
{disc(.18848,5, .24315,2,1,1); }
else
if(Scale == "100nm")
{disc(.01347,5, .0756,4,.15225,2,1,1);}
else
{disc(.08429,5,.13,2,1,1)};
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You have to use the decide module. It will serve the purpose. Work with ARENA V15 and above in case you do not have a commercial version. It allows upto 300 items, unlike earlier versions which are limited to 150 items
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In my opinion dynamical system is part of prediction science , but i know prediction is only interested for logical mathematical reformulation , Now my question is : Since dynamical system investigate about futur results for any phenomena according to known data and prediction science is also seek about futur mathematical logic , then What is the difference between prediction science and dynamical system ?
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Modeling is part of prediction science. Dynamical system investigate about futur results for any phenomena according to known data. I think this is what you mean by prediction science.
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Hello all,
I am aware of "formal methods" as it is used in computer science, to make sure our specifications are mathematically and logically sound (before we put them into a specific language). Isn't there a tool or form of notation so we can do that while we construct legitimate hypotheses for psychological research?
It is common knowledge that the construction of natural language questions (surveys about thinking and behavior), can be very questionable when it comes to construct validity. For example, did I cover everything? Am I even asking the right question? Trial and error without a formal test of our logic doesn't seem very efficient.
Does anyone know of a formal notation or research tool that lets us test natural language questions for how "sound" they are? After all, whether computer code (if/then we can do this) or human language (if/then we can assume that), it's pretty much the same logic. Thanks for any advice!
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I think you will need to address the use of formal methods in the society of engineering before you can use the methods more broadly. I am familiar with these methods, and I am quite skeptical of their application as you suggest.
However, I do believe that there is merit to the development of mathematics in the description of social behaviors on the underlying biophysical basis. A step beyond the statistical analysis and game theory so far.
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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?
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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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Knowledge representation may be constructed as an attempt to formally capture and describe human sensory and perceptual data. But is knowledge representation via ontologies etc., anything more than the application of logic and mathematics?
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This is an excellent question. In addition to the excellent answers already given by @Dejenie A. Lakew and @Peter Samuels, there is a bit more to add.
By way of starting an answer to this question, consider the following threads (that includes this thread):
Logic and a wee bit of mathematics (set theory) dominate knowledge representation in
Mathematics has more importance in knowledge representation in
More to the point, it is apparent that knowledge representation is not merely an application of logic and mathematics. Instead, logic and mathematics provide a concise language as a means of expressing knowledge, which is something quite different from logic and mathematics.
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It is true that during the “one-to-one correspondence” operation between Real Number Set and Natural Number Set, after the elements in Natural Number Set have been finished up, the elements in Real Number Set are still a lot (infinite) remained:
1, The elements in real number set are never-to-be-finished, endless, limitless------Real Number Set is really infinite!
2, The elements in natural number set are sure-to-be-finished, ended, limited-------- Natural Number Set is actually finite?!
There are still some other proofs of “one-to-one correspondence” operation between the two sets telling us a fact that the elements in many infinite sets are sure-to-be-finished, ended, limited and they are actually finite！
A typical tool and technique is Cantor's Power Set Theorem: all the elements in any infinite set can be prove “sure-to-be-finished, ended, limited and they are actually finite” in front of its own Power Set-------because during the “one-to-one correspondence” operation between the original set and its power set, after the elements in the original set have been finished up （finite）, the elements in its own Power Set are still a lot (infinite) remained!
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The question contradicts the fundamental principle of thought process and one of the basic axioms of ontology, something either "is" or " it is not", but not both. Based on the definition of a set being finite or infinite, if a set is finite then it can not be infinite and if it is infinite then it can not be finite. No finite set that is equivalent to any of its proper subsets and hence can not be infinite. Besides, the set of natural numbers do not end or finished as you have said (if that is what you meant)
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Following eight unavoidable suspended conundrums in set theory have been troubling us human for a long time：
1.      Does the definition of each “infinite set” closely relate to “the elements’ nature, appearance and inter-relationship” ---------the very characteristics of the elements inside the very “infinite set”?
2.      If it does not at all and the elements in all different infinite sets are the same a heap of “infinite abstract stuff without any differences and relationship”, then how can we define and distinguish “different infinite set” and how can we believe that there may be quantity differences between “different infinite sets”?
3.      If it does and the elements in all different infinite sets are “the concrete carriers of ‘abstract infinite concept’ with differences and relationship” (unique characteristics), then how will these unique characteristics decide the quantity differences between “different infinite sets”?
4.      Are “infinite sets” in present set theory “actual infinite sets” or “potential infinite sets”?
5.      Are infinite elements in infinite sets “actual infinite many” or “potential infinite many”? If they are “actual infinite many”, how can we conduct the quantitative cognitions to them; and if they are “potential infinite many”, how can we conduct the quantitative cognitions to them?
6.      What kind of mathematical tool of “one-to-one correspondence” is? When we conduct the quantitative cognitions to different infinite sets with “one-to-one correspondence” tool, is it “one element corresponding to one element” or “many elements corresponding to one element” or “many elements corresponding to many elements”, is it “potential infinite many elements corresponding to potential infinite many elements” or “actual infinite many elements corresponding to actual infinite many elements” or “potential infinite many elements corresponding to actual infinite many elements”?
7.      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?
8.        What kind of mathematical tool of “limit theory” is? When we conduct the quantitative cognitions to different infinite sets, how can we use limit theory to analyze, manifest and treat those X--->0 elements’ number forms inside them (such as those X--->0 elements’ number forms in [0, 1] real number set)?
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@ Geng Ouyang
Dear Mr. Geng Ouyang,
if you don´t read my proof to Cantor, you will not understand that you are not on the right way. Nevertheless we are working at the same problem. But please read! The research of a problem is not to add in a post.
To find no one-to-one-correspondence´ (bijectivity (?)) for two infinite sets is not a proof to show they are different in number (of amount). See my proof to Cantor (the natural numbers are able to count a infinite number (of amount) of sets of infinite amount of number of elements). The Real Number Set could not be described as a one which has more´ than infinite amount of number of subsets with each of them has infinite amount of number of elements. This more´ is in quantity. But infinity is quality.
Upvote: I upvoted your last answer when it was empty. Now (with your later added content) I am not in the same assessment.
Yours, Peter Kepp
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Paradox is a term that is used but not fully understood. A mathematical definition of paradox has been given by fuzzy logic, T= F= 1/2 but this appears to be incorrect.
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Dear Ashok
You understood me correctly:
You wrote "From my reading of our discussions, it seems that the issue is about the art of concluding from incomplete information."
This is right. Because objectively occurring events in physical reality can have an indefinite number of characteristics, of which we generally can become aware of only a part.
If we are not careful (if we do not completely verify), we end up concluding with insufficient information to form an objective opinion (conclusion).
We can analyze only what we allow ourselves to become aware of.
We can also chose to be thoroughly methodical about a posteriori verification and confirmation of premises.
In summary, there is
1- Objective physical reality with its indefinite number of characteristics.
2- What we perceive of this objective physical reality
3- What we conclude from what we perceived of this objective physical reality, which can be possibly correct even without verifying premises (if we are lucky) of incorrect (partial, paradoxes, etc) or confirmed (if premises are completely verified and found to be true)
The important fact is that we are physiologically unable to directly analyze 1. We are physiologically able to analyze only 2.
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I was looking for examples of first order sentences written in the language of fields, true in Q (field of rational numbers) and C (field of complex numbers) but false in R (field of real numbers). I found the following recipe to construct such sentences. Let a be a statement true in C but false in R and let b be a statement true in Q but false in R. Then the statement z = a \/ b is of course true in Q and C, but false in R.
Using this method, I found the following z:=
(Ex x^2 = 2) ---> (Au Ev v^2 = u)
which formulated in english sounds as "If 2 has a square-root in the field, then all elements of the field have square roots in the field." Of course, in Q the premise is false, so the implication is true. In C both premise and conclusion are true, so the implication is true. In R, the premise is true and the conclusion false, so the implication is false. Bingo.
However, this example is just constructed and does not really contain too much mathematical enlightment. Do you know more interesting and more substantial (natural) examples? (from both logic and algebraic point of view)
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Something algebraic, implicitly talking about ordering:
"for every nonzero number x, x or -x is a square but not both."
This holds in R (it is essentially an axiom of real closed fields) but not in Q or C (x=2 is a counterexample for both). Now you can take the logical negation.
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The problem is explained in the attachment. Note that the Laplace distribution is a negative example to the problem, despite that its characteristic function is meromorphic in the whole  C-plane.
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The name  "Cauchy  distribution" in the original formulation was wrong (as a negative example). Now it is replaced by the appropriate Laplace distribution (i.e. both sided symmetric exponential pd). Indeed, then  the characteristic function \phi(w) = 1/(1 + w^2) after the substitution of iw  for  w  and taking the inverse values is a polynomial - which is NOT a characteristic function of a pd. on  R.
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What I am curious to know is that, whether all the problems we have with arithmetics, like G2, still holds, if we change the concepts and properties of negation
Or
It can be wipe away by changing the definition of consistency.
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Yes. It is known since 1935-1970 (a bunch of results more and more refining your question).
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A_Extended=[A 0 0 0 0 0;
I 0 0 0 0 0 ;
0 I 0 0 0 0;
0 0 I 0 0 0;
0 0 0 I 0 0;
0 0 0 0 I 0],
where I is identity matrix and 0 is zero matrix?
Assume that eigenvalues of A are known.
If this is a difficult problem, can we say anything about being Hurwitz of A_Extended given that A is Schur?
I really appreciate your answers.
Best regards,
Mehran
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Peter: There is a more polite way to respond to someone that is asking for help.
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I am looking for a function form with vertical asymptote at x=0, horizontal asymptote at y=y0>0 and an inflection point along with minimum in between. I know rational functions with third order polynomial both in numerator and denominator work. A sample graph is attached so you have an idea of the form I am looking for. The function form is (rational function)
y = y0*(1+a/x+b(1/x)^2+c(1/x)^3)
However, I am looking for a simpler function only with one or two adjustable parameters (I know y0 but a, b, and c are adjustable). I would like to have the function completely positive (y>0). The important thing is I would like to be able to find regions for adjustable parameters over which the function is strictly positive. Any comments or advice highly appreciated.
Thank you.
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You can take any function of the form
f(x) = y_0 + a/x^p - b/x^q with p > q and a, b > 0.
The most famous example is the Lennart-Jones Potential (https://en.wikipedia.org/wiki/Lennard-Jones_potential)
V(r) = \epsilon*( (r_{\min} /r)^12  - 2(r_{\min}/r)^6)
which is a model potential for the van der Waals force between neutral atoms. It has a minimum of -\epsilon. Just adding \epsilon makes it non negative.
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Theorems use to be falsifiable. For instance, the well-known Pythagorean one need not be true in non-euclidean geometries. Is there any scenario in which Gödel's theorems do not hold?
If they are true under all circumstances perhaps they are tautological statements. From tautologies we can only deduce tautologies, hence they are of no use.
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Gödel's Theorem says non-trivial things about non-trivial objects. It has a very exactly defined scope and cannot be taken away to other objects. There is an interpretation of Gödel's Theorem that I like most. It says that the theory of the structure (N, 0, 1, +, *) is undecidable. There is no algorithm (Turing Machine) able to input formal statements in the given language and to correctly answer true or false (according to the fact that they are true or false in the given structure) although there is always exactly one good answer. Seen this way, there are a lot of decidable theories (as the theory of the dense order without end-points or the theory of the field of complex numbers) and a lot of undecidable theories (like the theory of the field of rational numbers, of the ring of integers or the theory of the reals with addition, multiplication and the function sinus). So Gödel's Theorem is falsifiable and does not express a tautology, because not all complete formal theories act in the same way. Please note that for every new theory, the result (decidable or undecidable) must be proven again. One cannot directly apply Gödel.
We can also refer to more intime details of  Gödel's proof as the consistent existence of a truth predicate. Well, there are theories consistently containing a truth predicate, at least for existential statements:
or
while arithmetic does not define such a predicate. So also this aspect of Gödel's result is not trivial or tautologic.
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It is analogous to Marcus equation- where L is lembda to calculate
∆G^‡=L/4 [1-(∆G^0)/L]^2
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Dear researcher, your equation is solved elementarily if you change Y = 1 / X:
P^2 Y^2 - 2(P+2Q)Y + 1 = 0, Y_1,2 = (P + 2Q +- 2 (Q(P+Q))^(1/2)) / (P^2).
So,
X_1 = (P^2) /  (P + 2Q + 2 (Q(P+Q))^(1/2)),
X_2 =  (P^2) / (P + 2Q - 2 (Q(P+Q))^(1/2))
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I am facing the problem in random case,as we know that eigen values of hermitian matrices should be real,and in my case when diagonal elements are random of hamiltonian,is it always necessary that deminant of Ham ,symmetric matrix is equal to product of eigen values or not.
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Thanks ,BUT my question is that why determinant is not coming equal to product of eigen values after 20,000 iterations,although it comes after like 2000 iterations,even i have read that people opt for Lanchzos algorithm ,instead of simple EIG functions.
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Let me recall that, for a positive integer n, Cut(n) is the statement that, for each sequence of n-element sets, the union of all sets of this sequence is at most countable. Cut(fin) is the statement that countable unions of finite sets are at most countable. I am unable to deduce whether it is true in ZF that if Cut(n) holds for each positive integer n, then Cut(fin) also holds. Perhaps, there exists a mathematician who knows a model for ZF in which Cut(fin) fails and, simultaneously, Cut(n) holds for each positive integer n. I would be grateful for any helpful hint to give a satisfactory answer to my question. Regards, Eliza Wajch
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I know that CUT(Fin) is equivalent to AC_omega(Fin), that is, all countable families of finite sets have a choice function. I guess that AC_omega(fatorial of n) implies CUT(n) - there are at most fatorial of n enumerations of a finite set with at most n elements, so choosing one enumeration for each one of the finite sets it is possible to proceed with some well ordering argument. So I guess that a equivalent questions is: is it possible to have simultaneously AC_omega(n) for every n and the failure of AC_omega(Fin) ? However, I am not aware of a model with such properties. I hope my remark helps somehow.
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The answer for the first order logic is in many elementary logic books. I looked in Wikipedia already. Anything that is somewhat related to this question would be appreciated.
If that will help here is more what I am doing. I look at satisfaction of second order formulas in models with infinite well ordered universe with pairing function and some additional model elements like predicates, constants and function symbols.
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One can keep all 2nd order quantifiers to the left.
See Takeuti's book.
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In our present infinite related mathematics theory system, no “infinite things” can run away from “actual infinite” and “potential infinite”. So, are there “actual infinite many (big)” and “potential infinite many (big)” in our mathematics?
The exactly same question goes to infinitesimals: are there “actual infinite small (few)” and “potential infinite small (few)” in our mathematics?  --------- Take Un--->0 in Harmonic Series and dx --->0 in calculus for example: (1) are they same infinitesimals under the same infinite concept with the same infinite definition? (2) are they “actual infinitesimals” or “potential infinitesimals”?
The thousand—year old suspended infinite related paradoxes tell us: something must be wrong in present classical “actual infinite” and “potential infinite” related philosophy and mathematics!
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Potential infinite was a concept defined by Aristotle. Actual infinite sets were introduced by Cantor in set theory, the mathemetical science of infinite.
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Are Natural Mathematical Processes a necessary component in the running of the Physical Universe?  Could the Universe operate without them?  For example, are mathematical processes essential in the addition of distances or displacements?
This question is an attempt at rewording of my earlier question asking whether mathematics is intrinsic in Nature.  That ealier question was rather open to interpretational debate.  I am hoping, this time, I more acutely address the point I  intend.
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What exactly do you envisage as mathematical processes, Steve? If I add 2 and 2 and get 4 I doubt my brain even computes. It just remembers the answer. An ensemble of photons going through 2 slits instantiate a physical process that for us requires complex numbers to describe. But surely nobody suggests that the photons or God are doing any complex arithmetic on the back of a celestial envelope.. What would a 'mathematical process' be?
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At least from Zeno’s time, most people (except Zeno with his creative “actual infinite--potential infinite” related paradox families) have been ignoring a fact that our whole present infinite related mathematics theory system has been based on “actual infinite--potential infinite”. The indefinable and confused “actual infinite” and “potential infinite” have been troubling us human for at least 2500 years with all kinds of debates, magics, paradoxes,… ------ the “2500-year-old huge black cloud of infinite related paradoxes over mathematics sky”.
Take Un--->0 in Harmonic Series and dx --->0 in calculus for example: (1) are they same infinitesimals under the same infinite concept with the same infinite definition? (2) are they “actual infinite small (few)” and “potential infinite small (few)”?
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There are both types of infinitesimals, plus Geometric infinitesimals! Actual infinitesimals are the Robinson's Nonstandard Analysis. On the other hand infinitesimals based on potential infinity is the Vopenka's Alternative Set Theory, which is based in turn on Nelson's Internal Set Theory (see e.g. Alain Robert Nonstandard Analysis)
Finally Geometric infinitesimals are the topos-theoretic ones, see e.g. Bell, J. L., A Primer of Infinitesimal Analysis. Cambridge 1998. Finaly there are Boolean-valued non-standard mathematics, (see, e.g. C. A. DROSSOS, G. MARKAKIS and P. L. THEODOROPOULOS,  B-FUZZY STOCHASTICS.)
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We have “big” and “small” in our science and the related numerical cognizing way to universal things around us （one of the mathematical cognizing ways）. So, we have “very big (such as 10001000)” and “very small (such as 1/10001000)”, “extremely big” and “extremely small”… in mathematics. When “infinite” came into our science and mathematics, we naturally and logically have “infinite big (infinities)” and “infinite small (infinitesimals)”.
Many people think we really have had many different mathematical definitions (given by Cantor) for infinite: infinity (infinite big) in set theory. So, when talk about the mathematical definition for “infinite”, people only think about “infinite big” but negate “infinite small”.
Should the mathematical definition for “infinite” cover both “infinite big (infinities)” and “infinite small (infinitesimals)” or only for the half: infinite big (infinities)?
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Dear Geng,
From a formal point of view, the construction of what you call ‘infinitely big’ and ‘infinitely small’ is different.
Cantor's infinite cardinal hierarchy arises from his attempt to generalise the idea of equinumerosity from the finite to the infinite. The limit construction in mathematics comes from the need to define in a mathematically concise manner a concept that was used in physical theorising (17th-18th  century Wallis, Leibniz, Mercator, Kepler, ) where 1/\infty, dx, partial derivatives and tendencies of series were used without proper definitions. (Note that the limit concept handled both things ‘series diverging to infinity’ and a proper use of the ‘infinitessimal dx’ in derivatives.) Finally the construction of hyperreals which provides another meaning to the notion of an infinitesimal, only arose in the context of first order logics and non-standard models of field theory.
One should also note the difference between the context in which a concept was proposed, and what it was later argued to represent. (Eg. hyperreals were not introduced to represent infinitesimals, although they are now thought to allow such representation and maybe much better than the limit construction. Similarly Cantor’s hierarchy of cardinalities was not introduced to represent the ‘infinitely big’, but was a side product of the attempt to formalise equinumerosity in purely set-theoretic terms).
There is, in principle, no good or bad in ‘infinities’ as there is no good or bad in any definition, as long as it is correctly and well defined. The question is rather, what you need them for (pragmatics of mathematics), what they allow you to understand (epistemology of mathematics) and whether you can do without it (ontology of mathematics). Your normative question of what should or should not be defined will have other meanings from these different perspectives and different answers, depending on what kind of stance you take on pragmatics, epistemology and/or ontology.
I hope this helps,
Best,
Eric
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The example I have in mind is a pair of integrals, over an inseparable domain, that do not exist unless weighted by functions in Banach space. But even if they are not, they can inter-substitute to imply existence of the Fourier transform and its inverse.
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I think that false physical models can combine and build together correct predictions. The big problem about this is that sometimes correct statements stay on false proofs. People find errors in the proofs, and then they claim that the statement itself is false. This is not always true. They proved only that this particular proof is false.
The situation is particularly embarassing as the last 150 years compose together the most active time in developping and concatenating physical theories.
Particularly, the statement E = mc^2 stays only on proofs which are literally false, but can still be correct. I don't know it to have been hardly fasified by experiments. Also "dark energy" and "dark matter" are almost sure inconsistent, but they build also some correct predictions together with other theories, which are not completely true. We need a fast and radical clean and clear action, or we run directly in a major crisis of science, faster than we believe.
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I am looking  for recursive formula for special kind of operations on power series which is called the substitution of one series into another. The main idea is the following. We have two power series:
\sum_{k=1}^\infty{b_k y^k}=\sum_{k=1}^\infty{c_k  x^k}
and
y=\sum_{k=1}^\infty{a_k x^k}
I would like to obtain general formula for c_k using known coefficients a_k and b_k.
In the handbook I.S. Gradshtejn, I.M.Ryzhik "Tables of integrals, series and products", AcadPress, 2007, page 17, the formula (0.315) gives only the first four coefficients of the power series which is the substitution of one series into another.
I would be very grateful for any tips or links to papers which can give the answer for my question.
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Dear Gregory,
Thank you for your clues. I have to admit that I have already read a little about Faà di Bruno's formula and it is really connected with my problem. On the other side I hope that my expected recurrence formula can be written in more clear (compact) form  than proposed by mentioned author (similar to other formulas in Ryzhik).
Best regards,
Radoslaw
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I would like to know a good text on non-standard models of Peano arithmetic. And also, any article about then. Thanks.
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Kossak and Schmerl: The structure of models of Peano Arithmetic.
Hajek and Putlak:  Metamathematics of First Order Arithmetic
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Could contradiction play a role in quantum systems, as part of the mechanism of measurement, forcing a single random outcome from the spectrum of possibilities?
All ideas are welcome, including outrageous ones.
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My idea is that contradiction may be the impetus that forces decision during quantum measurement.  Again referring to the donkey dilemma  -- the donkey that starved to death after being placed equidistant between two bails of hay.  The donkey cannot feed because no preference is possible.
If the donkey was forced to move forward by some strict contradiction behind him, forcing a decision on him, the contradiction would be imperative while the non-preference is no imperative at all.  What do you think about this kind of idea?
Steve.
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There have been two suspended questions challenging us human in present classical infinite related science system at least since Zeon’s time:
1, is it unavoidable that we mix (jump between) “potential infinite” and “actual infinite” whenever we cognize any infinite relating things?
2, do we treat potential infinite things or actual infinite things or jump between in mathematics quantitatively?
Can we have “potential infinite sets”? If yes, can anyone give an example of “potential infinite set”?
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Dear Professor Arno Gorgels and Professor Tadeusz Ostrowski，thank you.
Our science history tells us that for more than 2500 years, people have been “jumping up and down (on--off) as one wishes on the confusing actual infinite and potential infinite” and the “huge black cloud of infinite related paradoxes over mathematics sky” is produced by those defects disclosed by suspended Zeon’s Paradoxes family in present classical infinite related science system.
A revolution (in “infinite”, “infinite related number system”, “infinite related limit theory”) is on the way now.
In new infinite theory system, infinite is divided into “theoretical infinite and practical infinite” but not “potential infinite and actual infinite” and there will be no debates over potential and actual infinities any more.
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In arithmetics or algebras that cannot be completed, if any statement is logically independent of the axioms, is it also mathematically undecidable.  Are these concepts identical?
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The difficulty is the multiple use of the word undecidable'.   Here is some fairly standard notation.   1) A sentence phi is undecidable in  (or independent from) a theory T if both T union {phi} and T union {not phi} are consistent.
2) A theory T is decidable if its logical consequences form  a recursive set.
The easiest way to avoid confusion is to use only independent  from T and not undecidable in the first case.  1) is a property of a theory and sentence. 2) is a property of theory.
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The exercise I'm dealing with asks me to show that by adding S = K to the usual reduction rules for the SKI-calculus, one obtains an inconsistent equivalence. This must be done without using Böhm's theorem.
Now, I've found two terms (not combinators) M and N with the following properties:
M = x
N = y
M can be obtained from N by replacing one or more occurrences of S with K
From the rule S = K we thereby get that M = N, and hence that x = y. Which means that any term can be proved equal to any other.
Do you think such an answer could work? Actually, Böhm's theorem (as shown in the book I am studying) establishes that, for distinct combinators G and H, there is a combinator D such that DxyG = x and DxyH=y. So, I feel it would have been more appropriate to find a combinator D such that Dx1x2S = x_i and Dx1x2K = x_j with i , j = 1 , 2 and i ≠ j. But I have HUGE problems in finding combinators, so I have only found terms respectively reducing to one of the variables, and obtainable one from another by replacing S with K or vice versa.
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Yeah I see now there's a much simpler solution. But I wondered if I had "well done" the exercise. Actually, it seems to me it works. I found two terms, M and N, the first one reducing to an arbitrary x and the second one to an arbitrary y. Moreover, M can be obtained from N by replacing an occurrence of S with K and therefore, as soon as one admits S = K as a rule, it follows that M = N. Since M = x and N = y, by transitivity one should get x = y for every x and y.
Actually, M and N, as I built them, contain other terms which are in the end not relevant to respective reducibility to x and y. But that's should not be a problem. It should be like to have two terms with variables (x, y, z, w), possibly with z = w, respectively reducing to x and y.
That's how I did the exercise...
PS: inconsistent in the sense that one can trivialize the equality relation.
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I know that there is a model M for ZF such that. for an uncountable set S in this model and for every collection $\{ (X_s, d_s): s\in S\}$ of metric spaces in this model, their product $\prod_{s\in S} X_s$ in M is metrizable in M. In particular, for an uncountable set S in M, the product $\mathbb{R}^S$ is metrizable, however, I have not found this result in the literature so far. I would be grateful if you could tell me whether you have located  it in the literature. If your answer is YES,  please, tell me where I can find this result. I know how to prove the result.
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Since some researches have asked  me via e-mail how I deduce that R^J can be metrizable for an uncountable set J in a model for ZF, let me give to all concerned the following theorem:
Theorem. Suppose that  a non-empty set J is a countable union of finite sets and that $\{ (X_j, d_j): j\in J\}$ is a collection of (quasi)- metric spaces. If each $X_j$ is equipped with the topology induced by d_j, then the product $\prod_{j\in J} X_j$ is (quasi)-metrizable.
Now,  when M is a model for ZF+negation of CC(fin), there exists in M an uncountable set J such that J is a countable union of finite sets , so, for such a set J,   R^J is metrizable in M and, moreover, for example,  S^J is quasi-metrizable in M where S is the Sorgenfrey line. I was also surprised when I discovered these facts, They have looked strange in my opinion but they do not look strange now. They are relatively simple results.
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If there is a game with no rules, does it need a rule saying it has no rules?  Or can it just simply have none.
If a game has 3 rules, does it need another saying it has exactly 3?  Or can it just simply have 3?
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This question seems to same as the question I asked -- do we presume the structure of universe before defining it? https://www.researchgate.net/post/Do_we_always_presume_structure_of_universe_before_defining_it
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For example, assume two entities P and Q, where we are using 'proof-by-contradiction' to validate P (by using Q).
Something like:
• if P then Q;
• Not Q. Hence not P
IMO, one can only use such scenario, when P and Q are independent of each other existentially.
In other words, can one use proof-by-contradiction in cases where Q's existence is dependent on P's validity?
For example, Q exists if and only if P exists.
In such case does the proof-by-contradiction still hold valid?
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You must enlarge your question as if you could articulate ( kind of comparing ) the actual proposition with all the propositions like and axiomatic truthes and theorems system memories consist in so as "bugging" when these are processed or taken together. If you had already kept into memories only " truthes "  each  " new " proposition could be selected or rejected by this mean. But probably doing this you shut yourself to opportunities of radically " new " propositions that don't simply or immediately fit with older ones... It depends on you to work in a prehistoric or futur cyber-environnemental mood or with has been or promising concepts.
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Does it help us understand concepts of randomness and probability by considering the possibility that randomness may be involved in certain strange systems, where only one outcome is possible.  The motivation for this question is that a system with 1 possible outcome is the limiting case of systems with  ....6, 5, 4, 3, 2, .. possible outcomes.  OR  If a system with only one outcome has no algorithm for that outcome, but that outcome has probability one, could randomness in such a system be meaningful?
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Randomness is very helpful in systems with only one possible outcome. Example one: I (or you, the reader, or Schroedinger's Cat) will die. Example 2: A C14 atom will split. In both cases there is only one possible outcome, but we don't know when does it happen. Looked from outside, it is a matter of randomness, even if thee is only one possible outcome. And more important: this outcome is not a limit of a decreasing sequence of outcomes. It is intrinsically only one and intrinsically random.
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If the decision between two choices is to be made, but neither choice is preferred over the other, because there is perfect symmetry between the two, then no information separates the choices and the only decision that can be made is a random one.
What kind of imperative can force such a decision, and does such a decision resist the imperative?
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Bell inequalities compare two results obtained for the cases of probabilities (hidden variables, randomness) and amplitudes of probabilities (coherence). So far coherence wins over randomness,
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Quandles and racks are algebraic structures referred to in knot theory. Are there links to thorough introductions for researchers in how they are used in physics research?
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They are used in knot invariant generation.See the references below:
In statistical mecanics the Yang-Baxter Equation leads to a wealth of knot invariants and indeed they show up there: http://arxiv.org/pdf/math/0409202.pdf
If there is one candidate to what, I surmize, you are looking for it must be in that direction.
Hope this helps.
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PARITY is about whether a unary predicate of a structure has even numbers of elements in it.
If a kind of logic can define PARITY, then there is a formula of this logic so that:
PARITY return True on a structure iff this structure is a model of this formula.
We have known that logics with counting can easily define PARITY.
But what about others without counting?
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Right Peter. I just wanted to avoid such things as "there is a set" or "there is a function", so I proposed a first order formula containing given predicates and a function symbol. Better I make it formally here (for Arthur much  more than for Peter)
All finite models of the following axiom have even cardinality, and every set of even cardinality can be expanded to a model of the following axiom.
Language L = { A(.), f(.) } (one unary predicate and one unary function)
Axiom:
for all x, y [ (A(x) ---> not A(f(x))) and (not A(x) ---> A(f(x))) and (x =/= y ---> f(x) =/= f(y) ]
Let M be a finite model. The function f : M ---> M is injective. Let A \subset M be the set of realizations of A(x) and let M\ A be the set of realizations of not A(x).  So f(A) \subset M\A and f(M\A) \subset A. By injectivity of f follows | A | < = | M\A | from the first and | M \ A | < = | A | from the second. So | M \ A | = | A | , so |M| is even.
The theory is not complete! f can be an involution, or not. For example, M = {1, 2, 3, 4}, A = {1, 3} and f: 1 --> 2 --> 3 --> 4 --> 1 is not an involution; 1 --> 2 --> 1, 3 -->4 -->3 is an involution. The proposition "for all x f^2(x) = x"  is true in the second model but false in the first.
However every finite set with even cardinality can be expanded to such a model in many ways.
The problem might be that the negation of this axiom does not imply that the models are of odd cardinality. OK, I guess that at this point one would really need second order predicate logic! [If there do not exist subset A and function f such that the axiom is fulfilled, then the cardinality must be odd indeed]
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The following statement is well known in distributive lattices (D, 0, 1, ., +):
(1) Let F be a filter disjoint from an ideal I. Then there exists a prime filter F’ extending F and disjoint from I.
Usually the proof of (1) follows by an application of Zorn Lemma. But then the proof yields a stronger version:
(2) Let F be a filter disjoint from an ideal I. Then there exists a prime filter F’ extending F and disjoint from I and for every x not in F’ there exists y in F’ such that x.y is in I.
I am interested if (1) and (2) are equivalent (or not ) in ZF. Any references?
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No, they are not equivalent in ZF. Consider for example the case where I and F are trivial: I={0} and F={1}. Then (1) just says "there exists a prime filter" while (2) says "there exists a maximal filter". The existence of maximal filters in distributive lattices is equivalent to the axiom of choice AC while the existence of prime filters is equivalent to what is called BPI (boolean prime ideal theorem) which is known to be strictly weaker than AC, although not provable in ZF.
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Graph theory and mathematical logic, are both parts of Discrete Mathematics syllabus. Some logical equalities can be express by rooted trees.
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There is a strong connection between graph theory and mathematical logic. Several graph theoretical concepts can be definable in terms of first order logic and second order logic. One can use also the first order axioms on Interval functions on to define and characterize several graph families.
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I am looking for a proof such that:
Given a set of Horn clauses, show that there is a unit refutation from S if and only if S is unsatisfiable
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Resolution, thus unit resolution, is sound. So the "only if" part is immediate.
At the ground level, the "if" part is really easy, but the trick at the first order level is showing that "factoring" is not needed. JACM 21,4 is overkill, but it does this amongst other results.
Intuitively, a ground positive unit proof can be lifted to (a proof of) the first order clauses, of which it is a set of instances, in the usual way -- EXCEPT that a single ground step may correspond to a unit resolution involving a clause in which k literals have been factored together into one. But unification theory allows this one step on a factor  to be replaced by k (unit) steps on the original clause, resulting in the same resolvent produced by the one step with the factor.
(This is mostly intuition and not even a proof sketch.)
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Can The Physical Laws be regarded as a computational program that runs The Universe (Everything)?   If so, does the program need to generate The Physical Laws themselves?
This question is a followup on my earlier similar question referring to a computer simulation of The Universe. I give the link here below.
My purpose in asking these questions is to motivate the kind of physical theory that accepts that Physical Laws are part of The Universe, as opposed to standing outside it. And that rules governing The Universe must stem form The Universe itself. Otherwise, we should be asking: Where do the Physical Laws come from?
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Ellis
Why does there have to be a universe and stuff? It is possible that the only thing that exists is thought, and that the physical world is a construct of thought. By this I don't mean that thought actually constructs a physical universe in a material way, but that it constructs the concept of a physical universe to try to explain its own existence. This is a non dualistic view of reality, but equally valid as there is no way to determine the difference. I'm just taking the old brain in a jar dilemma a step further. The line between physics and meta-physics is blurry and constantly shifting. Logic is a branch of philosophy. In fact I would say that physics is a branch of philosophy as well since the empirical method came from it. It is only recently in history that we divide knowledge into increasingly more numerous separate boxes. Most likely, this is a result of the modern academic system and not a reflection of the nature of reality.
Shalender,
What makes you think physicists ignore the infinite nature of the universe? There are mathematical models for an infinite universe, both open and closed. Mathematics is perfectly capable of handling the concept of an infinite system. The concept of infinitely comes out of set theory after all. Can the universe be completely modeled by mathematics? I don't know, but just because we currently don't have a mathematical model that explains the entire universe doesn't mean that such a mathematical model doesn't exist. In fact I believe (correct me if I am wrong Steve) the question posed assumes that there does exist a mathematical set of laws that fully explains the universe. Also, what you describe with the lab experiment of a replica system is no different from how physicists model the physical world currently. It's called the scientific method. It is impossible to make a measurement of the entire universe, so we make local measurements and infer physical laws from them. These physical laws are then tested with measurements on increasingly larger portions of the universe (and by larger I also mean smaller, etc...) and modified if they are found invalid in their predictive power.
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If a computer program were to be written to simulate The Universe (Everything), would the simulation need to include the computer and program?  And if so, what process could boot this Universe from a totally off state?
My purpose in asking this question is to motivate the kind of physical theory that accepts that Physical Laws are part of The Universe, as opposed to standing outside it.  And that rules governing The Universe must stem form The Universe itself. Otherwise, we should be asking: Where do the Physical Laws come from?
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There are two theorems to confirm that you can't realize simulating a computer by this computer.
1) A Turing machine  T1 can't determine itself haft, so it can't simulatie itself.
2) Godel Second Uncompleteness Theorem says that a system T1 must be confirmed by a system  T2 larger than  T1 .
So there is not a limit to simulate T1,T2......
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I am studying the measurement problem in quantum mechanics.  I have a quantum theory based on a formal axiomatised arithmetic of scalars.  This is being considered under the disciplines of Mathematical Logic.  My arithmetic is a first-order theory and the tool I am using is first-order logic. I've had good success in finding logic in the quantum arithmetic, isomorphic with indeterminacy in quantum theory.  Specifically, I have identified, located and isolated logical independence in the wave packet for the free particle, just where indeterminacy shows up in experiments.  In order to make progress with collapse during measurement, I need to find mathematics that motivates a jump of the type, typified by the following.  Starting from the proposition: " there exists an x such that x=2 " -- jumping to the statement  " x=2 ".  Can anyone offer any ideas please.
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Just a word about the jump from "exists x such that x = 2" to "x = 2". The first formal statement contains a variable x which is bounded by a quantifier. So the statement is closed, it needs no supplimentary information to be understood. The second statement, "x=2", is open, because it speaks about an x that depends of the context. It may contain more information in the presence of (say) "x is the number of € Mary owns" and less Information in the presence of (say) "let x be a number". Other said, "x=2" is an open formula. How do you want to solve this difficulty? Very often, one adds constants to the language, so the second statement must be "c=2", and no "x=2". c=2 is a statement and not an open formula, it says something about two constants in the language.
The point is that the normal rules of proof in logic (the deduction rules) are Modus Ponens and Generalization. Generalization is sometimes expressed like this: from A(c) derive "exists x A(x)". What you say  is the other way around!... OK, you want this maybe only for "measurements" so formulae like "x = value", but this is still very non-standard. Question: would it suffice if you add a deduction rule like """ "exists x, x = value" implies "c=value" """"? I mean, just build an axiom containing what you need, and then study the resulted system: if it is consistent, sound, complete, etc...
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In a formal arithmetical system, axiomatised under the field axioms, the square root of minus one is logically independent of axioms.  This is proved using Soundness and Completeness Theorems together. This arithmetic is incomplete and is therefore subject to Gödel's Incompletenss Theorems. But can it be said that the logical independence of the square root of minus one, is a consequence of incompleteness?
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The word "incompleteness" has meanings which are slightly different in the situations exposed.
Meaning 1: The axioms of fields do not define a complete theory. (Proof: both R (real numbers) and C (complete numbers)  are fields. Consider the formal statement  S: " exists x such that x2= - 1". R is a model of  "not A" and C is a model of A. Therefore, A is Independent of the deductive closure of the axioms of fields, so this set of sentences (sometimes called also "the theory of fields") is not complete.
Meaning 2: Gödel's Incompleteness Theorem says that any recurrent sets of axioms that are true for the ring of integers Z and is consistent, cannot be complete. [please put the accent on the word "recurrent"] Other said, the theory of Z is undecidable. This means that there is no one algorithm able to decide the truth or falsity (over Z) of all formal statements in the language of Z. (The same can be said for the set of natural numbers R, as Gödel originally did.)
IN CONCLUSION: The word "incompleteness" has very clear meanings in the two situations given in the question, but the two situations are so different, that no connection can be done, as suggested in the question.
HOWEVER: In the book Tarski - Mostowski - Robinson, Undecidable Theories, ways are shown how to interpret Gödel's incompleteness in order to show that some uncomplete sets of statements are also undecidable. Using those techniques they show that the deductive closure of group axioms is undecidable (there is no algorithms permitting us to decide if a formal sentence in the language of groups follows from the group axioms alone) and do the same thing with different theories of rings. I believe that the theoty of fields is in the same situation: not complete and undecidable. Maybe some experts will clarify this point in their posts.
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Nowadays, the consistency of theories is not demanded and in alternative we search for relative consistency. In the future things may change. In particular, in the answer 70 and more easily in answer 76, it was proved that set theory is consistent as a result of a relative consistency. There were published several datasets proving the consistency of set theory. In the last times it was publshed a paper in a journal, without success, since there is some inertia concerning the acceptance of the consistency of NFU set theory. It can be said that NFU set theory is consistent as the result of a relative consistency: since Peano arithmetic is consistent than NFU is consistent too. By a similar argument it can be prooved that set theory is consistent too: since NFU set theory is consistent then set theory is consistent. Thus, set theory is consistent, and since the related proof can be turned finite then we also prooved the Hilbert's Program, that was refered in many books on proof theory. There is an extension of set theory, the MK set theory, which is a joint foundation of set theory and category theory, two well known foundations of mathematics.  Once again a paper by myself with title "Conssitency of Set Theory" was rejected without a valid reason. This agrees with an answer given by me 26 days ago. With set theory consistent we can replace the use of models to prove the independence of axioms  (as did by Goedel and Cohen) by deduction in set theory.
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Zermello Fraenkel (ZF) and Zermello Fraenkel with Choice (ZFC) are only proposed Axiom systems. If they prove unconsistent, that does not mean that there are no sets, that there is no mathematics anymore, etc. This only means that a tentative of first order foundation by a system of schemes of axioms was unconsistent, and that the problem is open again. However, it is very likely that a new system of axioms set theory will most probably have the same problem as ZF: we will prove immediately that it is impossible to prove its consistency. So finally we will maybe adopt a position near to that of Bourbaki, in spite of its so called ignorance, see
To sum up: both if ZF is consistent or not, this question remains a never ending story by ist own and intimate nature. However, inspite of the possible instability of such an axiomatic construction, basic objects of mathematics like N, Q, R, C continue to exist and mathematics continue to be done. An incosistency of a given system of axioms does not touch the object for which it was standing. It touches only the problem of finding the right system of axioms for the given object.
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I am a research scholar, working on transformer fault diagnosis and classification. Can anybody tell me how to calculate training time, in seconds, of transformers, using Back Propagation Neural Network, Fuzzy Logic and Adaptive Neuro-Fuzzy Inference System?
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If you would like to estimate the run time of any programs in Matlab environment, you could write “tic” on the beginning of the program and “toc” at the end.
tic : start  a stopwatch timer
toc: stop  a stopwatch timer
enjoy
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Like Aristotle's four causes, what are the different types of reasons?
I'm thinking there are one's based on sentiments, passionate and computational. That's as far as I've gotten. I'd love to hear comments additional ones. Also, here are just a couple stumbling blocks I've come across:
For example, what kind of reasoning did Gödel use to liberate himself from the very limitations which he argued were characteristic of sufficiently developed computational devices?
Or what about poor Gretchen in Goethe's Faust. She commits infanticide, yet she is the symbol of innocence throughout the epic. What kind of reasoning did she use to commit her action? Here we get into the very importance of law with the different types of reasoning.
Can anyone else elaborate or add to this small list of the different types of reasoning?
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There are four basic forms of logic: deductive, inductive, abductive and metaphoric inference. In deduction inference leads fro true propositions to true propositions. In induction we can infer from cases to generalizations, which get conformation from premisses. In abduction one can infer causes from effect, thus going backward from "conclusions to premisses". Abduction is also called diagnosis. In metaphoric inference  we transfer knowledge from  one area to another area, say we study economy in terms of evolution theory.
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It was true that mathematics was done in argumentation and discourse or rhetoric in ancient times. The 6 volumes of Euclid’s elements have no symbols in it to describe behaviors of properties at all except for the geometric objects. The symbols of arithmetic: =, +, -, X, ÷ were created in the 15th and 16th centuries which most people hard to believe it - you heard me write. The equality sign “=” and “+,-“ appeared in writing in 1575, the multiplication symbol “X “ was created in 1631, and the division sign “ ÷” was created in 1659. It will be to the contrary of the beliefs of most people as to how recent the creations of these symbols were.
It is because of lack of symbols that mathematics was not developed as fast as it has been after the times where symbols were introduced and representations, writing expressions and algebraic manipulations were made handy, enjoyable and easy.
These things made way to the progress of mathematics in to a galaxy – to become a galaxy of mathematics. What is your take on this issue and your expertise on the chronology of symbol creations and the advances mathematics made because of this?
http://Notation,%20notation,%20notation%20%20a%20brief%20history%20of%20mathematical%20symbols%20%20%20Joseph%20Mazur%20%20%20Science%20%20%20theguardian.com.htm
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Leibniz was the master of symbol creation!  He created symbols that packaged meaning,  helped cognition, stimulated generalization, and eased manipulation.  He thought about them with care before committing to their use.  William Oughtred invented hundreds of new symbols, but hardly any of them are still in use.  Goes to show that willy-nilly made symbols don't have a good survival rate, for good reasons.
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Yes, I know. They say, the 2nd order logic is more expressive, but it is really hard to me to see why. If we have a domain X, why can't we define the domain X' = X u 2^X and for elements of x in X' define predicates:
SET(x)
ELEMENT(x)
BELONGS_TO(x, y) - undefined (or false) when ELEMENT(y)
etc.
Now, we can express sentences about subsets of X in the 1st-order logic!
Similarly we can define FUNCTION(x), etc. and... we can express all 2nd-order sentences in the 1st order logic!
I'm obviously overlooking something, but what actually? Where have I made a mistake?
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An essential difference between first order and higher order logic is Löwenheim-Skolem. In first order logic, a consistent theory which have infinite models will have models in every cardinality greater or equal with ist language. So, if Zermello-Fraenkel is consistent, there are countable models of the set theory! Inside the model, one has all infinite cardinalities - but looked from outside, they are all countable sets! In first order logic, if a consistent theory has an infinite model of some cardinality, there will be models in all bigger cardinalities. This is not so in higher order logics. In second order logic one can write down the condition to have a linearly ordered field, real closed and complete. There is only one field enjoying those properties: the field of real numbers R. So there are no countable models and no models of bigger cardinality - just one model of cardinality 2^\omega. In the same way one can define archimedean fields in second order logic, but not in first order logic. It is much more difficult to make model theory in second order logic, and almost impossible to make things like nonstandard Analysis...
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If we have an n by n matrix called A. How do we know if there is an inverse matrix A^-1 such that the product A * A^-1 is the n by n identity matrix?
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Yes Jose Vegas is right. Infact, what you should have if det(A) is non-zero A.Inv(A) = Inv(A).A = I the identity matrix. However, for large value of n it is difficult to find det(A). If you apply, Gauss elimination method, then during elimintion process t some point your diagonal element becomes zero can not be made non-zero by elementary row exchange then the matrix is singular and the inverse does not exist.
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Please mention the name of some books and prerequisites?
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Smullyan has a great chapter in the edited work "The Mathematical Sciences" (1969) called "The Continuum Hypothesis". Another really accessible article is "The Continuum Problem" (2002) in American Mathematical Monthly vol 109, pp 286-297. Finally, Erwin Schrodinger gives a nice discussion in his article "Causality and Wave Mechanics" on pp 1060 in J R Newman's massive "The World of Mathematics" (1956), vol 2.
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Peirce’s law, (((P-->Q)-->P)-->P), is a law of the Pure Conditional Sentential Logic.
Q1: Who proved that it is not intuitionistically provable? Q1.1: What is the proof?
Q2: Who proved that it is not classically provable using only the standard -->Intelim rules? Q2.1: What is the proof?
Q3: What “meanings” have been suggested for it? How could it have been discovered? Why should anyone conjecture it?
Q4: What good is it? What results in logic use it?
Q5: Hassan Masoud observed that its unprovability using only the standard -->Intelim rules suggests that the standard -->Intelim rules should not be regarded as “determining the meaning" of the connective --> Has this or related observations been discussed in print?
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Instantiating Q by Falsity gives you
(\neg A -> A) -> A
and thus
\neg\neg A -> A
which axiomatizes classical propositional logic on top of intuitionistic propositional logic.
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It would appear that the probability of randomly selecting a specific element from a Transfinite Set would be zero. However, NOT making a selection at all, would also yield a zero probability. Thus the contradiction. Furthermore, this can be compounded further if we exercise the above consideration on countable and uncountable Transfinite Sets. Are some 'zeros' less than others?
ADDENDUM (Feb. 2016): Let us consider the above posed question as selection from a transfinite set of points on two Lines perpendicular or parallel:  L1 and L2 .  Now let us re-evaluate this in terms of Purdy's Conjecture, to wit.,  Suppose that n points are to be chosen on line L and another n points on line M. If L and M are perpendicular or parallel, then the points can be chosen so that the number of distinct distances determined is bounded by a constant multiply of n, but otherwise the number is much larger.
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I like your approach about transferring our discourse to a Probability Space, but I'd like to stay on point here (pun intended). Of course a Transfinite Set has no supremum per se, but the heuristic of my question is how a probability selection is affected by the nature of how we chose to characterize the constituents of the line under analysis. In this context, it is my suspicion that a lush flourishing of insights might be gained when we consider not the Reals (or Hyper-reals, for that matter), but the underlying subtext of denumerability and nondenumerability within the universe of discourse and the sets under consideration. A tangentially related aspect of this may be some probabilistic schemes in testing Semantic Security in a hypothetically unbounded encryption algorithm.
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I would like to know what books are considered, from the personal experience point of view, more suitable for teaching Theorem Proving, when teaching this topic for Computer Science students. Usually it is a very complicated issue for students, specificaly those studing in first or second courses.
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The Art of Proof :Basic Training for Deeper Mathematics by Mathias Beck and Ross Geoghegan
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Recall that recursive definitions are those defining statements involving the defined object.
The defined object can occur either explicitly or implicitly in the recursive definition.
For instance, the definition.
x = positive number that coincides with x^2.
This recursive definition determines de number 1. The defined object x occurs implicitly. By contrast, in the following
definition
x = positive number that coincides with its powers.
contains x implicitly; because of, in this definition, the expression "its powers" is equivalent to x^1, x^2, x^3 ….
Another example is
x = set of all sets.
This definition is also recursive, because defines x as a set, therefore and the expression "all sets" involves x.
In fact, a recursive definition fits into the pattern
x = p(x)
where p(x) stands for any predicate involving x.
In general, recursive definitions are stated in several metamathematical topics without proving its consistence.
Nevertheless, if the pattern x = p(x) is regarded as an equation instead of a definition, then it is required
to show that such an equation has at least one solution. Why do not require also the existence proof in recursive definitions?
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@Ivan Suskov Suppose the domain is the positive integers and f(∅)=∅, and otherwise f(S) is the subset obtained when the smallest integer in S is deleted. In this case, would your "least solution" be ∅?
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I'm currently writing a short essay on Formal Verification and Program Derivation, and I would like to include an example of an algorithm that seems to be correct under a good number of test cases, but fails for very specific ones. The idea is to present the importance of a logical, deductive approach to infer the properties of an algorithm, in order to avoid such mistakes.
I can't quite think of a good example, though, because I'd like something simple that a layman can understand, but all the examples I came up with are related to NP-complete graph theory problems (usually an algorithm that is correct for a subset of the inputs, but not to the general problem).
Could someone help me think of a simpler example?
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A folklore example is swapping the value of two (integer) references without using a local variables. Take a programming language with call-by-reference and the following function:
swap(&int a, &int b) {
a := a - b; // value of a is a0 - b0
b := a + b; // value of b is a0 - b0 + b0 = a0
a := b - a; // value of a is a0 - a0 + b0
}
Assume the value of a is a pointer to the value a0 and b to b0. Then after a call to swap(a, b) the value pointed to by a and b are swapped ... except if a and b both point to the same memory address, in which case they point to the value 0.
This is more an example of the problems related to reasonning about programs in the presence of pointers but I think that it qualifies as an example of tricky algorithm.
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For analysis purposes I need the coefficients of matrix powers C=A**m. Given a (n,n) matrix A with coefficients a(i,j) one can define recursive equations for the coefficients c(i,j):
For 1 <= i,j <= n and 1<=p<=m
c(p,i,j) = a(i,j) for p=1
c(p,i,j) = Sum(1<=k<=n, c(1,i,k) * c(p-1,k,j)) for p>1
but neither the sign of the result nor a closed formula can be seen immediately.
Does anyone have an idea to resolve the recursion?
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Are you aware of the standard solution to your problem: Diagonalize your matrix, compute the m-th power (trivial for a diagonal matrix), transform back? O course, diagonalization does not work for all types of matrices. Sometimes playing arround with Mathematica helps.