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Right now, in 2022, we can read with perfect understanding mathematical articles and books
written a century ago. It is indeed remarkable how the way we do mathematics has stabilised.
The difference between the mathematics of 1922 and 2022 is small compared to that between the mathematics of 1922 and 1822.
Looking beyond classical ZFC-based mathematics, a tremendous amount of effort has been put
into formalising all areas of mathematics within the framework of program-language implementations (for instance Coq, Agda) of the univalent extension of dependent type theory (homotopy type theory).
But Coq and Agda are complex programs which depend on other programs (OCaml and Haskell) and frameworks (for instance operating systems and C libraries) to function. In the future if we have new CPU architectures then
Coq and Agda would have to be compiled again. OCaml and Haskell would have to be compiled again.
Both software and operating systems are rapidly changing and have always been so. What is here today is deprecated tomorrow.
My question is: what guarantee do we have that the huge libraries of the current formal mathematics projects in Agda, Coq or other languages will still be relevant or even "runnable" (for instance type-checkable) without having to resort to emulators and computer archaeology 10, 20, 50 or 100 years from now ?
10 years from now will Agda be backwards compatible enough to still recognise
current Agda files ?
Have there been any organised efforts to guarantee permanent backward compatibility for all future versions of Agda and Coq ? Or OCaml and Haskell ?
Perhaps the formal mathematics project should be carried out within a meta-programing language, a simpler more abstract framework (with a uniform syntax) comprehensible at once to logicians, mathematicians and programers and which can be converted automatically into the latest version of Agda or Coq ?
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I just encountered a notification about this article on Mathematical Proofs and the value of Proof Assistants, https://www.ams.org/journals/notices/202401/rnoti-p79.pdf
I resonate most with the positions by Lamport (author of LaTex and a Turing Award Laureate) and Laurence Paulson (author of ML for the Working Programmer and working very much in proof assistants). I think it will be clear that the situation about the presentation of proofs and incorporation of such proofs in mathematical publication is still very much up in the air.
I also did a ResearchGate search on "Proof Assistant" and the fire-hose of articles confirmed my view that this is yet stabilizing, although there are extensive favorite approaches.
Here is the above paper's abstract:
“A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into peril. Now may be the time to reconcile theory and practice, i.e., precision and intuition, through the advent of computer proof assistants. This used to be a topic for experts in specialized communities. However, mathematical proofs have become increasingly sophisticated, stretching the boundaries of what is humanly comprehensible, so that leading mathematicians have asked for formal verification of their proofs. At the same time, major theorems in mathematics have recently been computer-verified by people from outside of these communities, even by beginning students. This article investigates the different definitions of a proof, the gap between them, and possibilities to build bridges. It is written as a polemic or a collage by different members of the communities in mathematics and computer science at different stages of their careers, challenging well-known preconceptions and exploring new perspectives.”
There is already an objection to this material on the list where I found it. The objection is to this statement: "This puts the unique status of mathematics as exact science into peril.” That statement disturbs me too, but maybe not for the same reason.
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How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?
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A list of open problrems for hpmotopy type theory is presented in HISTORIC.
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People may have different names for ”the infinite related very small numerical things (infinitesimals)”, it doesn’t matter what they are called, they are there in our mathematics, but what are their positions as numbers or non-numbers or something else theoretically and practically, ontologically and formally?
The newly discovered modern Harmonic Series Paradox is one of family members of ancient Zeno’s Paradox, it discloses relentlessly a fact that we human still don’t know what infinitesimals are!
This problem has close relationship with whole fundamental part of infinite related area in our mathematics:
1, theoretical and practical infinite concept system
2, theoretical and practical infinite related number system
3, theoretical and practical infinite related number treating system
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Dear Mr. Louis Kauffman , thank you very much for your idea!
You admit there is really “The large difference between theory and practice” in our infinite related mathematics.
The problem is, I believe (in fact I am on the way doing something) that just because mathematics is human’s, we can try some way to fill up “The large difference between theory and practice” in our infinite related mathematics to solve many suspended “infinite related paradoxes” (paradox syndrome complex) in present mathematical analysis and set theory---------the "thousands--year old huge black clouds of infinite related paradox families over the sky of present classical mathematical analysis".
Regards!
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The problem disclosed by Zeno’s Paradox is still there and the exactly same idea is still working well. Let’s see one of the modern versions of Zeno’s Paradox
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)
Such an antique proof (given by Oresme in about 1360), though very elementary, can still be found in many current higher mathematical books written in all kinds of languages.
Here, with limit theory and technique, we see a “strict mathematically proven” modern version of ancient Zeno’s Paradox:
1, in Harmonic Series, we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 100000000000000000000 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  (such as Un’ >10000000000000000000000000) ;
2, the “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 100000000000000000000 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.
Lacking the systematic cognition to “infinitesimal”, no one in the world now can answer following question scientifically and this is the very reason for many “suspended infinite related paradox families” in present classical infinite related mathematics:
Are “dx--->0 infinitesimal” in calculus and “Un--->0 infinitesimal” in Harmonic Series the same things? If ”yes”, why we have totally different operations on them? If ”no”, what are the differences and how to treat them differently and why?
--------Could anyone tells how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the first Un’ >10000000000000000000000000, how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the second Un’ >10000000000000000000000000, how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the third Un’ >10000000000000000000000000?
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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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There are two “reasonable” limit operations in convergent- divergent proof of Harmonious Series:
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)
(1) During the whole process in dealing with infinite substances (infinitesimals) in limit calculations, no one dare to say “let them be zero or get the limit”. So, the infinitesimals in the calculating operations would never be too small to be out of the calculations and the calculations dealing with infinitesimals would be carried our forever. This situation has been existing in mathematics since antiquity------ those items of Un--->0 never be 0 all the time and Harmonious Series is divergent, so we can produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Un--->0 items in Harmonious Series and change an infinitely decreasing Harmonious 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. Here, with limit theory and technique, we see a “strict mathematically proven” modern version of ancient Zeno’s Paradox.
(2) During the process in dealing with infinite substances (infinitesimals) in limit calculations, someone suddenly cries “let them be zero or get the limit”. So, all in a sudden the infinitesimals in the calculations become too small to stay inside the calculations, they should disappear from (be out of) any limit calculation formulas immediately. This situation has been existing in mathematics since antiquity-------those items of Un--->0 must be 0 from some time and Harmonious Series is not divergent, so we cannot produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or…from infinite Un--->0 items in Harmonious Series. But if it is convergent, another paradox appears.
But when and to which should or should not people treat infinitesimals appearing in infinite numeral cognitions that way?
Does limit theory need basic theory, what is it?
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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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Many people agree that we are really able to produce infinite items each bigger than 1/2, or 100, or 1000000, or 100000000000000000000000000000,…by the brackets-placing rule with limit theory from Un--->0 items in Harmonic Series and change Harmonic Series into an infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000).
Can anyone prove how many Un--->0 items in Harmonic Series we use to produce the first three (just the first three!) positive constants of 100000000000000000000000000000----------how many Un--->0 items for the first 100000000000000000000000000000, how many Un--->0 items for the second 100000000000000000000000000000, how many Un--->0 items for the third 100000000000000000000000000000?
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The two statements
a) "the sum of the series is infinite" 
and
b) "the partial sums of the series will reach eventually N, 2N, 3N,... kN for a given N and for any positive integer k"
are equivalent, for any series of positive numbers.
That simply comes from the definition of the limit being +infinity.
So of course, if you assume a) you can prove b) and if you assume b) you can prove a).
It turns out that since the harmonic series is divergent, then both statements a) and b) are true when applied to the harmonic series.
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(1)  How  can we   find    the partial sum    of   n1000  instantly ?  
(2)  is  there  is  a  simple method to find  partial sum  of the  sequence  f(n)  ?
(3)  Any general method   to compute partial   sum of sequence   ?
(4)  What is the  value  of   Method , if we have   good  approximation  for all differentable  sequence   ?
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@ Juan Weisz  
That  is more than that   .  It  is   a little secret   have little secret     of  analytic continuity   and discreetness  of  integers   
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Following six unavoidable suspended conundrums by newly discovered Harmonic Series Paradox have been troubling us human since antiquity:
1, what are those Un--->0 items in Harmonic Series, are they infinitesimals?
2, if they are not, what are they and what is infinitesimal?
3, if they are infinitesimals, are they potential infinitesimals or actual infinitesimals, why?
4, if they are potential infinitesimals, how to have numerical cognition to them; if they are actual infinitesimals, how to have numerical cognition to them?
5, what kind of mathematical tool “limit theory” is, does it treat “potential infinite related mathematical contents (number forms)” or “actual infinite related mathematical contents (number forms)”?
6,can we really produce infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000000,…by that “Brackets-Placing Artifact” with limit theory from Un--->0 items in Harmonic Series and change Harmonic Series into an infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000)?
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First a word about the Tortoise race, the key to understanding is that for any practical physical measurement Achilles and the Tortoise have stopped moving, which is the only way that Achilles does not pass the Tortoise.
Next it should be said that infinitesimals only occur as abstractions. Infinite terms can be compared to each other but not to finite terms.
For every paradox there is a false assumption. Examples the Tortoise continues to run, and the harmonic oscillator continues to vibrate in higher modes, both of which are not physically true, although unreal abstractions can be made for them.
I've learned that people who dwell on paradox are happy with an unsolved riddle, and disappointed to learn there is an answer.
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So many people say “infinite thing is not a number”, but we are able to take very freely all the infinite things (such as infinities and infinitesimals) as numbers in the field of analysis and set.
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The question is not well posed. "Infinities" normally refer to cardinalities. It is known that if A is infinite set, then P(A) is a strictly bigger infinite set. This yields infinitely many infinite cardinalities.
Infinitesimals refer to elements of a non-archimedean ordered field, which are in absolute value smaller than every positive rational number. Their multiplicative inverses are "infinitely big elements" - but not infinities. For n in N, if e is infinitesimal then ne is infinitesimal; respectively if B is infinitely big, then nB is infinitely big. So there are infinitely many infinitesimals and infinite many infinitely big elements in the non-archimedean ordered field.
In both cases there is really nothing special. Just a little bit of mathematics.
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There are three different ways to deal with “infinite things” quantitatively in our actual limit operating behaviors:
(1) During the whole process in dealing with infinite substances (such as infinitesimals) in limit calculations, no one dare to say “let them be zero or get the limit”, because they are believed to have infinite property and they are existing forever for the whole of infinite. So, the infinitesimals in the calculating operations would never be too small to be out of the calculations and the calculations dealing with infinitesimals would be carried out forever. This situation have been existing in mathematics since antiquity, only in Zeno’s time he discovered something unscientific and created his Zeno’s Paradox to criticize this phenomenon bravely and sharply, the newly discovered Harmonic Paradox is its “modern strict mathematical proof version”.
(2) During the process in dealing with infinite substances (such as infinitesimals) in limit calculations, someone suddenly cries “let them be zero or get the limit”, because they are suddenly believed to lost infinite property and they only exist for the portion of infinite. So, all in a sudden the infinitesimals in the calculations become too small to stay inside the calculations, they should disappear from (be out of) any limit calculation formulas immediately. This situation have been existing in mathematics since antiquity, only in Berkeley’s time he discovered something unscientific and created his Berkeley’s Paradox to criticize this phenomenon bravely and sharply; our studies have proved that The Second Mathematical Crises can never be solved in present “potential infinite-actual infinite” related classical science system.
(3)Whenever infinite substances (such as infinitesimals) appear in limit calculations, someone cries “let them be zero or get the limit”. So, no calculations are needed and there would be no mathematical analysis at all. But we all know it is never allowed. Why?
The problem is: no one in the world now can tell self-justificationly why should or should not people treat “infinite things” appearing in limit calculations that way and how to understand and explain our behaviors relating to the “process and results of limit calculating operations”!
Are there really fatal defects disclosed by those suspended paradox families in limit theory related quantitative cognition ideas and behaviors to “infinite things”?
The exactly same situation happens in “non-standard analysis”.
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Interesting question, but let us first differentiate between infinitely large and infinitesimally small, and even zero.  In my opinion, zero is most suitable for countable items (apples), or items that either exist or not (loads).  Infinitesimally small (or very small for practical purposes) always exists.  One question is where we should draw the line, such as what should we consider to be zero, say, 1.0e-7 or 1.0e-14, or 1.0e-217.  Any large scale software programmer always runs into this question, and the answer is data, theory, and software dependent, so I will not dwell on that.  The more important question, and I think the one that you wish to pose, is how we should treat our expressions such that nothing blows up when we are getting near zero, and so that we get some intelligent results.  One way of doing this, again in my experience, is by considering what our tolerances should be for a zero of a function, and usually we have a good idea of this based on the problem we are trying to solve.  By applying this tolerance to our denominator (as applicable :) ), we can make sure that we are never dividing by zero.  As the tolerance gets closer to zero, and as the rest of the denominator (selected wisely, so it does reflect movement towards zero) gets to zero, the effect will be a smooth convergence to a non-zero number, one that is as reasonable as your expression and as your tolerance.  Now, as for infinitely large, that also exists (as in resonance, or with nonlinear springs, at least in theory), but I have no tolerance for that (pun intended), except for avoidance of dividing by zero, as noted above.  Good luck with those very small values.
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Our science history tells that we human have been arguing and debating on the nature and definition of “potential infinite and actual infinite” for at least 2500 years since Zeno’ time, and there are still no results nowadays. So, when facing “infinite things” in present classical infinite related science theory system, no one is sure whether they are “potential infinite things” or “actual infinite things” and are not sure how to treat them scientifically------- many suspended old and new infinite related paradox families are produced (such as 2500-year old Zeno’s Achilles--Turtle Race Paradox and so many modern versions of Zeno’s Achilles--Turtle Race Paradox, Russell's Paradox and so many modern versions of Russell's Paradox, …)by the confusing of “potential infinite and actual infinite” and these paradox family members are surely unsolvable in present classical infinite related mathematics where they were produced and nourished.
One may say “look at the history of mathematics and don’t mind the arguing and debating on the nature and definition of potential infinite and actual infinite, our mathematics goes well without the definition of infinite, just close one eye and open another eye in the field of infinite and we are used to this since Zeno’ time”.
But the facts are: more and more suspended infinite related paradox families are produced by the “confusing of potential infinite and actual infinite” such as infinitesimal relating paradoxes in analysis, infinity relating paradoxes in set theory, both infinitesimal and infinity relating paradoxes in the ideas and skills of “Cantor’s diagonal-contradictory proofs and the conclusion on “Real Number Set has more elements than Natural Number Set (infinite elements in Real Number Set is more infinite than that in Natural Number Set-------Infinite R is more infinite than Infinite N)”.
Can we really just close one eye and open another eye in the field of infinite and force us be used to the fundamental defects in our mathematics?
Trying our best to have scientific foundation for our mathematics is a must we should shoulder sooner or later beyond our will.
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Dear Dr. Chris Cook, thank you for your quantitative ideas on the “infinite”.
In fact, there are 3 problems make us unable to discuss “quantitative infinite” scientifically in our present “potential infinite-actual infinite” related classical science system:
1, no one knows whether the “infinite” mentioned in any “quantitative infinite” question “potential infinite” or “actual infinite” because no one and nothing can run away from the confusing of “potential infinite” and “actual infinite”;
2, even some one decides that the “infinite” is sure to be “potential infinite” or sure to be “actual infinite”, no one in the world now is able to define scientifically what “potential infinite” is and “actual infinite” is-------there have been no scientific definitions for both “potential infinite” and “actual infinite” at least since Zeno’s time 2500 years ago, so we have endless debates on them;
3, the “infinite” is “non- number number”--------- on the one hand, we have to cry out loudly and firmly first in mouth that “infinite small and infinite big are not numbers” while these “non-number infinite numbers” are participating all kinds of calculations with numbers; on the other hand, we tell in mind without any hesitation: “forget what have been just said in mouth for those very infinite small and infinite big, they are in fact numbers other wise no practical numerical operations can be done”. Most of us have to play the role of noble ministers in Andersen’s fairy tale The Emperor's New Clothes, compiling wonderful sayings to persuade everyone believe that “this emperor is a special one and he has no clothes on while has clothes on”--------the infinite small and infinite big participants in number calculations are “numbers while non-numbers”, “finite things while infinite things”, “potential infinite things while actual infinite things”,….
Yours,
Geng
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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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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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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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The choice to be made is similar to the choice between set theory and category theory.
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@Tadeusz Ostrowski: In her [Mathematics] palace there are rooms for all theories.
In addition to each of the incisive (and interesting) answers, already given, the answer given by @Tadeusz Ostrowski (and @George Stoica) makes perfect sense: rather than choosing between set theory and topos theory as a foundation of mathematics, we can view Mathematics as a form of Hilbert Hotel (there is always room for one more guest).   In terms of this thread, a new guest would be a fundamentally important theory in the foundations of mathematics.
I myself favour axiomatic set theory, especially if we start considering various forms of topology.   
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There have been two opposite phenomenon about the definition for infinite in present classical infinite related science system: (1) plenty of different mathematical definitions for infinite and (2) no mathematical definition for infinite.
On the one hand, we really have had many different mathematical definitions for just half of the infinite: infinities (infinite bigs) in set theory such as “super (higher, more, less…) infinite, super (higher, more, less…) unbounded, super (higher, more, less…) unlimited, super (higher, more, less…) endless,…”, the plenty of non-self-justification “half definition” for infinite which is nothing to do with infinitesimals strongly prove that we human in fact are unable to know how to have mathematical definition for “infinite” at all------it is impossible to have mathematical definition for “infinite” in present classical infinite related science system.
On the other hand, we really have had many different mathematical definitions for another half of the infinite: non-number infinitesimal variables, actual infinitesimals,… in standard and non-standard analysis, but those suspended infinitesimals related paradoxes (the 2500—year suspended black cloud over mathematical sky), especially the newly discovered Harmonic Series Paradox in present classical infinite related mathematics make us have to admit that till now we human in fact are unable to know how to have mathematical definition for “infinite” at all------it is impossible to have mathematical definition for “infinite” in present classical infinite related science system.
It is the time for us human to work at the integration of mathematical definition for “infinite” now?!
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One should bring in two definitions for infinite. One should be called relative infinite with a different symbol, and the other the absolute infinite.When we connect the abstract truth to the physical truth we can cognize, there is always a convenience factor varying  to the stretch of human imagination. Nice question to dwell on to stumble on the mysterious to unlock further secrets.
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Limit theory is a very useful cognizing tool in human science. It treats “infinite number forms”. But its foundation is empty:
1, what kind of “infinite number forms” it treats-------potential infinite number or actual infinite number or missing ups or …?
2, in present mathematical number system, there are no “infinite number forms” at all.
We should do something to make up the defects in the foundation of limit theory.
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On one hand it is a trivial and well known fact that we will catch the Turtle if we are faster than the Turtle. On the other hand Analysis proves that we catch the Turtle. However, the limit is NOT 1 + 1/2 + 1/3 + .... In the case of Achiles and the Turtle the limit is 1 + 1/10 + 1/100 + ... + 1/10^n + ... in the plausible case that Achiles is 10 times faster than the Turtle. Attention: this is not the harmonic series. This is the geometric series! So again: our experience shows that we catch up the Turtle, and Analysis proves that we catch up the Turtle. So where is Zeno's paradox? 
Zeno's paradox is that it takes infinite time to READ an infinite series, even if the series converges. This is normal because words like "1/10^n" take longer and longer time to be read (with n increasing), while they refer to smaller and smaller time intervals in the infinitesimal process of catching up the Turtle. Happily we introduced shorthand notations for all recursive functions. We can describe the function by the algorithm that produces it, and every algorithm has a finite description. So finally what will be faster: to read """Sum_{n \geq 1} 1/10^n"""" or to catch up the Turtle? This depends only on how long is the distance I let the Turtle in its advantage. 
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It is believed that there are the bijection relationships between Infinite Natural Number Set and Infinite Rational Number Set, but following simple story tells us that Infinite Rational Number Set has far more elements than that of Infinite Natural Number Set:
The elements of a tiny portion of rational numbers from Infinite Rational Number Set (the sub set : 0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) map and use up (bijective) all the numbers in Infinite Natural Number Set (0,1, 2, 3, 4, 5, 6, …, n …); so,infinite rational numbers (at least 2,3,4,5,6,…n,…) from Infinite Rational Number Set are left in the “one—to—one element mapping between Infinite Rational Number Set and Infinite Natural Number Set (not the integer set )------- Infinite Rational Number Set has infinite more elements than Infinite Natural Number Set.
This is the truth of a one-to-one corresponding operation and its result between two infinite sets: Infinite Rational Number Set and Infinite Natural Number Set. This is the business just between the elements’ quantity of two infinite sets and it can be nothing to do with the term of “proper subset, CARDINAL NUMBER, DENUMERABLE or INDENUMERABLE”.
Can we have many different bijection operations (proofs) with different one-to-one corresponding results between two infinite sets? If we can, what operation and conclusion should people choose in front of two opposite results, why?”
Such a question needs to be thought deeply: there are indeed all kinds of different infinite sets in mathematics, but what on earth make infinite sets different?
There is only one answer: unique elements contained in different infinite sets -------the characteristics of their special properties, special conditions of existence, special forms, special relationships as well as very special quantitative meaning! However, studies have shown that, due to the lack of the whole “carriers’ theory” in the foundation of present classical infinite theory, it is impossible for mathematicians to study and cognize those unique characteristics of elements operationally and theoretically in present classical set theory. So, it is impossible to carry out effectively the quantitative cognitions to the elements in various different infinite set scientifically -------a newly constructed Quantum Mathematics.
The article《On the Quantitative Cognitions to “Infinite Things” (IX) ------- "The Infinite Carrier Gene”, "The Infinite Carrier Measure" And "Quantum Mathematics”》 has been up loaded onto RG introducing the working ideas. https://www.researchgate.net/publication/344722827_On_the_Quantitative_Cognitions_to_Infinite_Things_IX_---------_The_Infinite_Carrier_Gene_The_Infinite_Carrier_Measure_And_Quantum_Mathematics
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Dear Geng
Lets use your own reasoning in a different way:
Just take a tiny portion of the Natural Number Set (2,4,6, ... ,2n, ...) and they map very well on the set of all Natural Numbers  (1,2,3,...,n,...). So a lot of natural numbers are left behind in this one -to-one mapping (2n onto n) from the Natural Numbers onto the Natural Numbers, Hence your conclusion should be: the Natural Number Set has far more elements than the Natural Number set. Think about this.
Best regards, Joseph.
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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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No question here...
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Joel, take the following: 
A is the set of the numbers that are integers, greater than 13, and even. 
B is the set of the numbers that are integers, greater than 13, even, and may be expressed as a product involving the factor 3. 
A={14, 16, 18, ..}
B={18, ...}
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How can we develop a list of micro foundations for organizational capabilities? is such a list different for different capabilities?
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Interesting differentiation - I appreciate the clarification. There seems to be an interesting parallel between foundations at various levels. Starting with foundations for structures - they must be inter-connected to be useful. Similarly, on another level, assets must be interconnected to be effective (must be able to draw on funds, access organizational efficiency, etc.). And, for mental models, the same is also true.
There is a study using "integrative complexity" that shows a correlation between managerial success and the level of structure of their mental models. That is to say, when we have mental models whose concepts are more complex and interconnected, we have a better understanding of the world, and are therefore able to make better decisions and be more successful.
reference provided in the attached paper.
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This paper provides a simplified exposition (no real analysis) of the economic theory presented in the second part of my 1999 book, Axiomatic Theory of Economics.  It makes no mention of the first part of my book about the foundations of economics.  In this question we will discuss my three-term system of formal logic, specifically with comparison to the attached paper by Steve Faulkner, which was posted in reply to another question that I recently asked.
In Section 4.3 of my 1999 book I write:
“The great crack in the foundation of mainstream logic where first-sense and third-sense truth are confused has been resolved.  Whenever mainstream logic speaks of affirmation they refer to phenomena having been observed that conform to a definition (truth in the first sense) and whenever they speak of negation they refer to the impossibility of phenomena conforming to a definition (falsity in the third sense).  The three senses of truth must be strictly separated…”
The three sense of truth are defined in an earlier section of my book, but suffice it to say that I was not bothered so much by the paradoxes that Gödel addressed but by the fact that, if p is impossible, the statement “some p are q” is false while the statement “all p are q” is true.  This is absurd.  If I told you that all red-headed Eskimos can foresee the future, a logician would have to admit that, within his science, this statement is true.  But everybody else would denounce me as a lunatic:  Eskimos do not have red hair and nobody – regardless of the color of their hair – can foresee the future.  The logical truth value of my statement will not inspire anybody to travel to Alaska to find Sibyl the Eskimo with her flaming red hair.
“A new system of formal logic will now be introduced.  The three terms of this system of logic are P for possible, I for impossible and M for maybe (similar to Zen Buddhism’s mu.)  Following are eleven logical relations concerning the definitions p and q.  These statements are followed by a truth table which shows, in each of the four situations with which one could be presented when observing phenomena’s conformance to p and q, whether the statement affirms its possibility, its impossibility, or says nothing about that situation.”
While I do not have space here to print the entire list of eleven logical relations, I will print the truth table for “p is possible unless q is possible” to give a taste of what I am doing:
Do phenomena conform to definition p?                T   T   F   F
Do phenomena conform to definition q?                T   F   T   F
p is possible unless q is possible                            I   P   M  M
I then use an example from Willard Quine’s Methods of Logic (p. 196) to illustrate how my method works:
Premises:
1)  The guard searched all who entered the building except those who were accompanied by members of the firm.
2)  Some of Fiorecchio’s men entered the building unaccompanied by anyone else [unaccompanied by non-Fioreccio men].
3)  The guard searched none of Fioreccio’s men.
Using my system, by filling in a truth table with P (possible), I (impossible) and M (maybe), we can quickly determine if the statement, “Some of Fioreccio’s men are members of the firm” is proven.  There is no room to print this here, but it is a sixteen-column truth table with four rows of P, I or M for each of the three premises and the relation, “people who work for Fioreccio.”  Below this is another row labeled “result.”
“Now, filling in an I wherever we see one, a P wherever we see one that is not dominated by an I, and an M only where no statement is made either way, we get the result.”
This is in contrast to Dr. Quine’s method (p. 199), which only proves or disproves one statement at a time.  I write:
“From this result [the three-term truth table] one can test the truth of any conclusion one is interested in…  If we were interested in knowing whether the statement ‘All of Fiorecchio’s men entered the building unaccompanied by non-Fiorecchio men’ is implied by the premises, we would need [elaborate what is needed that we do not know] so the conclusion is not proven; it is a maybe.  This is a more insightful ‘maybe’ than we had before analysis, however, as we now know where our investigation must lead.”
REFERENCES
Quine, W.V.  1982.  Methods of Logic.  Cambridge, MA:  Harvard University Press
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Victor, I am both a mathematician and a philosopher, and I work a lot with Logic, which is my preferred part. 
I have just studied Godel's proof of God and you can see the result of that in the book On Religion, which is found on my profile here.
I think that the aim of Logic is not confusing people, but making things easier and nicer instead. 
Priest, like you, loved formalizing anything. 
The problem with that is that you end up working in the abstraction, but you then want to come back to reality in the end and, whilst the first move could be acceptable, if things were ever done properly, the second move is usually impossible to to accept. 
Basically, I have written a lot about this by now: We can fit a reduced universe into a large one, but there are lots of things that escape logic in human language/discourse and the reverse move is not usually OK.
The sentences you wrote are easily dealt with without us changing them into symbols and, therefore, the need to do that does not exist. We are not simplifying if doing that either, we are actually complicating and probably missing the point, so that that is also not advisable. 
What Prof. Faulkner says is true and it actually occurred to me as I read your writing, but in a much more informal way. 
If you read more of my work, you will understand why this concern you seem to be having is actually equivocated for Mathematics. Creating new systems of logic, as Priest does, is fun, but may also be completely useless activity. 
Apart from learning the concepts involved in the modelling of the systems and applying those, there is nothing to it. 
If you want them to be useful, you must make sure you find a situation for which we feel the necessity of coding things. In second place, as you have probably noticed, you should also check if the system already exists. 
Remember that we must improve things with them, not confuse people even more or lead them to mistake in reasoning.
Priest has a book with almost all non-classical systems and you could actually use his book, since it works like an encyclopedia, to find out if a system already exists or not. 
I think he finished this book in 2000, after he went through it with me. 
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Whether exist an paraconsistent version of PA which can prove own paraconsistency?
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Paraconsistency cannot be proven if you refer to the ontological paraconsistency, as I state on my work about translation, and you can find a sample here. 
Ontological Paraconsistency is just a fraud: It is based on mistaken assumptions. We are actually referring to distinct perspectives, not the same one, when defending it. The foundations are equivocated.
Paraconsistency, as in a logical system, does exist, as an abstraction. 
I assume you cannot prove the bubble exists when you are inside of the bubble, as I say on my work about Film Blowing Modelling, so that you should not be able to prove paraconsistency with the own paraconsistency. You would have to always be at least one level above, is it not? 
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“Potential infinite” and “actual infinite” are really there in our mathematics and science, but it seems very difficult to understand and express these two concepts clearly and logically ever since.
In present infinite related science and mathematics, people have been admitting the being of “potential infinite, actual infinite” concepts, unable to deny their qualitative differences and the important roles they play in the foundation of present classical infinite theory system and, unable to deny that the present infinite related classical mathematics is basing on present classical infinite theory system. The fact is: on the one hand, present classical mathematics can not avoid the constraining of “potential infinite--actual infinite” concepts and their relating “potential infinite mathematical things --actual infinite mathematical things”; on the other hand, no clear definitions for these two concepts of “potential infinite--actual infinite” and their relating “potential infinite mathematical things --actual infinite mathematical things” have been constructed since antiquity , thus naturally lead to following two unavoidable fatal defects in present classical mathematics:
(1)it is impossible (unable) to understand theoretically what the important basic concepts of “potential infinite, actual infinite” and their relating “potential infinite mathematical things --actual infinite mathematical things” are. So, in many “qualitative cognizing activities on infinite relating mathematical things (infinite relating number forms)” in present classical mathematicis, many people actually don’t know or even deny the being of “potential infinite, actual infinite” concepts and their relating “potential infinite mathematical things --actual infinite mathematical things”--------the “qualitative cognizing defects on infinite relating mathematical things (infinite relating number forms)”. 
(2)it is impossible (unable) to understand operationally what kind of relationship among the important basic concepts of “potential infinite, actual infinite”, their relating “potential infinite mathematical things --actual infinite mathematical things” and all the “infinite mathematical things as well as their quantitative cognizing operations” are. So, in many “quantitative cognizing activities on infinite things (infinite mathematical things)” in present classical mathematics, many people have been unable to know whether the infinite relating mathematical things being treated are “potential infinite mathematical things” or “actual infinite number forms”, no one has been able to avoid the confusing of “potential infinite mathematical things” and “actual infinite mathematical things”, no one has been able to know whether or not treating “potential infinite number forms” or “actual infinite number forms” with the same way or different ways. What is more, many people actually don’t know or even deny the being of “potential infinite mathematical things” and “actual infinite mathematical things”--------the “quantitative cognizing defects on infinite relating mathematical things (infinite relating number forms)”.
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It is possible that peoples of different linguistic backgrounds conceptualise infinity differently due to peculiarities of their languages. I came across an interesting article about this problem on RG . The paper was written by Dong-Joong Kim, Joan Ferrini-Mundy and Anna Sfard; it is titled: "How does language impact the learning of mathematics? Comparison of English and Korean speaking university students’ discourses on infinity".
The article claims that "It was found that in spite of the comparable levels of mathematical performance, there was, indeed, a visible dissimilarity between mathematical discourses on infinity of Korean- and English speaking students. In general, whereas no group could pride itself on a well-developed mathematical discourse on infinity, the mathematical discourse of the English speakers, just like their colloquial discourse, was predominantly processual, whereas the Korean speaking students’ talk on infinity was more structural and, in an admittedly superficial way, closer to the formal mathematical discourse." 
This could explain the inconclusive results of many interesting discussions initiated by Geng. You can find the article on Anna Sfard ResearchGate site. 
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Since “infinite” concept came into our science, the “infinite” related concepts and theories such as “potential infinite”, “actual infinite”, “countable infinite”, “uncountable infinite”, “infinity”, “infinitesimal”, “infinite set”, “variables” were introduced; still, some other mathematicians (such as G. Kantor and A. Robinson) have tried to develop some different “infinite” theories specially (only) for set theory or analysis …. The question of “What is potential infinite and actual infinite?” has been analyzed, discussed and debated and this situation is sure to be “endless” in present classical science theory frame--------our science history strongly proved!
Our studies prove that when facing and treating the “infinite related beings” in present cluttered, unsystematic classical “infinite” theory system, we are unavoidable to meet following two unexplainable arguments: (1) what on earth are “infinite”, “potential infinite”, “actual infinite”, “higher infinite”, “lower infinite”, “the ‘infinite’ of more infinite”, …? Can we really have many different definitions for “infinite”? Are different definitions for “infinite” the same mathematical things in our science? Why? (2) What kind of “infinite related number forms” should we have to demonstrate and cognize so many different “infinites”? Can we use just one kind of “infinite number form” forthem (several “infinite related number forms” in Harmonic Series Paradox is a typical example)? Why? Cardinality, continuum hypothesis and non-standard analysis theories help nothing here.
Our science history since Zeno’s time tells us clearly: there are serious fundamental defects in present infinite related classical science theory system-------both in philosophy and mathematics. Our science history since Zeno’s time also proved that not matter how we have tried, all the paradoxes and troubles produced by present infinite related classical science theory system are impossible to be solved (unsolvable) inside this very system itself.
For some small defects, the diminutive mendings are very much ok; but for the serious fundamental defects, those diminutive mendings do not only of no help but produce more troubles------errors plus other errors. So, the challenge is: to be or not to be staying in the foundation of present infinite related classical science theory frame.
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Actual and potential infinity are not different "kinds of infinity". They have to do with different ways of dealing with infinity. When proving a theorem by natural induction -- say: Gauss' sum formula for finite arithmetic progressions-- you have been dealing with potential infinity: the formula is verified for each natural number. When using the set of all natural numbers in set theory, you are effectively using an infinite object. You are now dealing with actual infinity.
Cantor's infinite cardinal numbers can indeed be seen as degrees of infinity. Using expressions like "more infinite" for this make it sound ridiculous, but that is just playing with words. That the cardinality of the real number system (the continuum) is larger than the cardinality of the natural number system corresponds with a fundamental fact that one cannot run all reals in a counting process (Cantor's diagonal argument). It also demonstrates that a majority of reals are transcendental. Similarly, the power set of a set S outperforms the original set S by a famous argument of Cantor, which may have inspired the Russell paradox. This (so-called) paradox was just a warning that Cantor's paradise doesn't come for free; it can only be entered with a suitable axiom system for set theory.
I don't see the exact problem of this question. You can adhere to finitary mathematics if you wish, rejecting (actual) infinity. If not, you must face the fact that the concept of infinity comes with refinements which should not be approached with naive intuition. It's a technical thing.
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The following proof (given by Oresme in about 1360), very elementary but important, can 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)
Each operation in this proof is really unassailable within present science theory system. But, it is right with present modern limit theory and technology applied in this proof that we meet a “strict mathematical proven” modern version of Ancient Zeno’s Achilles--Turtle Race Paradox[1-3]: the “brackets-placing rule" decided by limit theory in this proof corresponds to Achilles in Zeno’s Paradox and the infinite items in Harmonic Series corresponds to those steps of the tortoise in the Paradox. So, not matter how fast Achilles can run and how long the distance Achilles has run in “Achilles--Turtle Race”, there would be infinite Turtle steps awaiting for Achilles to chase and endless distance for him to cover, so it is of cause impossible for Achilles to catch up with the Turtle; while in this acknowledged modern divergent proof of Harmonic Series, not matter how big the number will be gained by the “brackets-placing rule" (such as Un’ >100000000000000000) and how many items in Harmonic Series are consumed in the number getting process by the “brackets-placing rule", there will still be infinite Un--->0 items in Harmonic Series awaiting for the “brackets-placing rule" to produce infinite items each bigger than any positive constants, so people can really produce infinite items each bigger than 1/2, or 100, or 1000000, or 100000000000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 100000000000000000), “strictly proving” that Harmonic Series is divergent. In so doing, the conclusion of “infinite numbers each bigger than any positive constants” can be produced from the Un--->0 items in Harmonic Series by brackets-placing rule and Harmonic Series is divergent" has been confirmed as a truth and a unimpeachable basic theory in our science (mathematics) while “the statement of Achilles will never catch up with the Turtle in the race” in Ancient Zeno’s Achilles--Turtle Race Paradox has been confirmed as a “strict mathematical proven” truth and a unimpeachable theorem------it would be Great Zeno’s Theorem but not Suspended Zeno’s Paradox! ?
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I'm quite sure that they can give nice answers to any unsolved problems we have to deal with, including universe or quantum. I'm really not a specialist for it but I feel that's a really good field of research to find an issue to our fundamental questions...
the best to you all
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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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I.  On Fri Oct 5 23:35:03 EDT 2007 Finnur Larusson  wrote
         I confirm that Larusson "proof"  under corrections can be formalized in ZFC.
II. On Sun Oct 7 14:12:37 EDT 2007 Timothy Y. Chow  wrote:
"In order to deduce "ZFC is inconsistent" from "ZFC |- ~con(ZFC)" one needs
something more than the consistency of ZFC, e.g., that ZFC has an
omega-model (i.e., a model in which the integers are the standard
integers).
To put it another way, why should we "believe" a statement just because
there's a ZFC-proof of it?  It's clear that if ZFC is inconsistent, then
we *won't* "believe" ZFC-proofs.  What's slightly more subtle is that the
mere consistency of ZFC isn't quite enough to get us to believe
arithmetical theorems of ZFC; we must also believe that these arithmetical
theorems are asserting something about the standard naturals.  It is
"conceivable" that ZFC might be consistent but that the only models it has
are those in which the integers are nonstandard, in which case we might
not "believe" an arithmetical statement such as "ZFC is inconsistent" even
if there is a ZFC-proof of it.
So you need to replace your initial statement that "we assume throughout
that ZFC is consistent" to "we assume throughout that ZFC has an
omega-model"; then you should see that your "paradox" dissipates.".
J.Foukzon.Remark1. Let Mst  be  an  omega-model  of  ZFC  and let ZFC[Mst]  be a  ZFC with a quantifiers bounded on model MstThen easy to see that  Larusson  "paradox"  valid inside  ZFC[Mst
III.  On Wed Oct 10 14:12:46 EDT 2007 Richard Heck wrote:
Or, more directly, what you need is reflection for ZFC: Bew_{ZFC}(A) -->
A. And that of course is not available in ZFC, by L"ob's theorem.
J.Foukzon.Remark2 However such reflection is .available in ZFC[Mst] by standard interpretation of Bew_{ZFC}(A) in omega-model  Mst    
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Remark4. However note thatthe condition of the existence standard model of ZFC    which has been requested  by T.Chow
is not necessary. It well known that any nonstandard model of FA (named MNst )  is necessary contains such minimal standard model of FA.see Theorem 3.1
Order-types of models of Peano arithmetic
MNst=N+AZ
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When we want to find out the gap between upper bound and lower bound, we can use Gap to see how much upper bound deviates from lower bound.
What is exactly the meaning of deviate in here?
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The gap you are referring to is called the "duality gap", or "Lagrangian duality gap". There is an optimal value of your original ("primal") problem, and an optimal value of your Lagrangian dual problem, and the duality gap is the difference between these two values. When your primal (original) problem is a minimization then the Lagrangian dual problem is a maximization problem, and by weak duality every feasible point in the primal problem has an objective value which is higher (or at least never lower) than any feasible point has with respect to the Lagrangian dual objective function in the corresponding Lagrangian dual problem. 
The duality gap is zero for convex problems under a regularity condition, but for general non-convex problems, including integer programs, the duality gap typically is positive. This is then a serious problem when trying to "translate" a dual optimum into a primal optimum. I have co-written a paper in the journal Operations Research, for which you can find a late revision here: 
If you need the published paper I am sure you can find it. The paper shows precisely where the duality gap arises, by analyzing the global optimality conditions. It also describes ways to use these conditions in an algorithmic fashion. That has, sofar, not been a great success algorithmically, but at least it provides a precise statement, which possibly could be used in your case.
So - in order to find what the gap is you need to find a better way to solve the primal problem - your heuristic may fail to find an optimum, and even if it does you will not know that it has. 
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In pure mathematics there is no absolute truth [Stabler]; we invent rules then see what they prove or see what is consistent with them.  So in physics, what kind of truths are we looking for? Are we looking for absolute truths in physics?
(Note that the premise of my question immediately contradicts itself -- saying there is not absolute truth is an absolute statement. Maybe it is not a mathematical statement, I'm not sure.  Apologies for this mess.)
Ref:  Edward Russell Stabler, An introduction to mathematical thought, Addison-Wesley Publishing Company Inc., Reading Massachusetts USA, 1948.
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Akira, Welcome to physics!  ha ha!  This is the reason we have made essentially no progress on fundamental problems in the last 60+ years.  Low bars and big egos and the sensible people skulk out of the room before the sparks fly.  
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I trying to extend some results that hod true for reflexive spaces, and I'm wondering whether this assumption could be too strong in a general LCS.
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Results on the reflexivity of locally convex spaces can be found in "Topological Vector Spaces", by H. H. Schaefer, Springer,1971.
@James Peters: yes indeed, most of the basic results in functional analysis had already been proved before the period 1970-1980. The book of R. B. Holmes is a very good one on this subject. It contains also convex (concave) optimization problems.
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As we may know, in the standard approach to the $q$-calculus there are two types of $q$-exponential functions $e_{q}=\sum_{n=0}^{\infty}\frac{z^{n}}{\left[ n\right]_{q}!}$ and $E_{q}(z)=\sum_{n=0}^{\infty}\frac{q^{\frac{1}{2}n\left(n-1\right) }z^{n}}{\left[ n\right]_{q}!}$. Based on these $q$-exponential functions, also, some new functions are defined whose most of their properties are similar to the exponential function in calculus. Despite having interesting properties similar to the exponential function in calculus, for none of them the above property holds. So, can we define a new $q$-exponential function with similar characteristics to the exponential function in calculus which satisfies the aforesaid property?Any help is appreciated.
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Unfortunately, if you are interested in continuous solutions of your functional equation F(x+y)=F(x)F(y), then the exponential functions exp(a x) are the only answer, see the links below for details (note that if F satisfies the above equation, f=ln(F) satisfies the Cauchy functional equation f(x+y)=f(x)+f(y)):
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People believe that 0 (zero) has played different rolls as a kind of special form of number (substantiality) in our science, such as:
(1) A kind of reference ------- generator, middle, neutral, beginner, origin, placeholder, marker,…,
(2) Absolutely non-existent ------- without numerical value meaning, the negation of being, objectively nothingness,
(3) Relatively non-existent ------- with numerical value meaning, subjectively nothingness, the approximate nothingness, the result of infinitesimal limit , ….
Lacking the systematic cognition to 0 (zero), no one in the world now can answer following question scientifically and this is the very reason for many “infinite related paradox families” in present classical infinite related mathematics:
Are “dx--->0 infinitesimal” in Calculus and “Un--->0 infinitesimal” in Harmonic Series the same things? If ”yes”, why we have totally different operations on them? If ”no”, what are the differences and how to treat them differently and why?
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Mihai's examples of temperature measurement and (more generally) of measurement by marking an initial point as "0" and by marking a second point as the unit "1" of measure, clearly illustrate the respective role of these numbers. This is highly compatible with a mathematical viewpoint, where "0" and "1" are simply the names given to numbers with are neutral for addition, respectively, multiplication. This terminology extends to all algebraic structures which can be considered as a "number system".
One might say that "0" and "1" are special numbers because they calibrate the system. In case of measurements of any kind, 0 and 1 each represent the outcome of some agreed default situation. As to "0", this default is often some kind of failure (zero distance, zero dollars on your US bank account). But "zero degrees Celcius" and  "zero-th floor" (downstairs, at least in British English) are of a different nature.
It seems that "0" is only occasionallly related with "non-existence" and even then, it is usually some kind of agreement.
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We focus on the “deep structural relationship” between “nonstandard one” and “standard one”. Let’s exam following facts:
1, as “monad of infinitesimals” has much to do with analysis; nonstandard analysis is much more a way of thinking about analysis, as a different analysis------simpler than standard one.
2, CONSERVATIVE is the nature and a must for Nonstandard Analysis or Nonstandard Mathematics, it is called a conservative extension of the standard one.
3, because of the “deep structural CONSERVATIVE”, the “provable” equivalence are guaranteed.
If there are “no defects” in the “standard one”, the “CONSERVATIVE guaranteed nonstandard” work would be really meaningful.
Now the problem is “nonstandard one” inherits all the fundamental defects disclosed by “infinite related paradoxes” from “standard one” since Zeno’s time 2500years ago------guaranteed by the “deep structural CONSERVATIVE” .
Theoretically and operationally, “nonstandard one” is exactly the same as those of “standard one” with suspended infinite related defects in nature.Simpler or not weights nothing here.
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Hi Akira, I think a firm theoretical bridge between discrete systems and continuous approximations to them may not be possible any time soon.  Again, if a physicist can find an excuse to treat a large discrete system (e.g., a container of gas) as a continuous entity, she will.  "Anything that gets answers."   As for supporting/refuting physical theories, only degrees of confidence are required to underpin belief.  I don't believe empirical data near as much as I believe a mathematical argument;  and the more complicated the argument, the more difficult it is for me to believe. 
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A transitive action of the automorphism group means that for any two nonzero elements there is an automorphism taking one element to another.
We're working on finite modules over finite rings.
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Ali.
For doubly-transitive permutation groups, G.M with regular normal subgroup M, M is always an elementary abelian p-group, for some prime p, so it is automatically a R0=GF(p)-module [finite field of order p], and G acts via R0-linear automorphisms. But actually, there will be a maximal field R=GF(q), where q=p^m, such that M is an R-module and G acts via R-module automorphisms. So really, what I said about classifying ALL doubly-transitive perm group actions is unnecessary: M is a finite-dimensional vector space over finite field R=GF(q) and conversely, if M is an n-dimensional vector space over R, there is a transitive subgroup of R-automorphisms: namely GL_n(R)!
Now, if you start with M which is an R1-module for some finite ring R1 (maybe not assumed an integral domain or, say, left-associative, or ..., so not necessarily a finite field, a priori) then, in any case, M is a finite abelian group under addition. But if there is a group G of R1-automorphisms of M that is transitive on M\{0}, then G is, in any case, a group of additive automorphisms of M and G.M is as above.
Thus, M is always an elementary abelian p-group and R1 <= R = GF(q), where R is the finite-field ring of G-endomorphisms.  But then R1 HAS to be commutative, associative etc., so is a finite field - a subfield of R.
So, the two things together give the answer to your question, I think:
the finite ring R HAS to be a finite field for the existence of a transitive group of automorphisms, but if R IS a finite field, so M is a finite dimensional vector space over it, then the group of R-module automorphisms is transitive on M\{0}.
[Note that for finite rings, a finite division ring is automatically commutative, so is a finite field, GF(q)].
Anyway, if you are interested in this sort of problem, I think it is a good idea to have a look at the theory on primitive and multiply-transitive finite permutation group actions, because that's where the answers are!
Best Regards, Mike Harrison
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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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“Infinite”, “potential infinite”, “actual infinite”, “potential infinitesimal”, “actual infinitesimal”, “potential infinite-big”, “actual infinite-big”, “Infinite related numbers”,…; What are they?
Some people insist we can we have only one definition of “infinite” in science but others argue that we can have many definitions of “infinite” with different natures in science (at least two: “potential infinite” and “actual infinite”), what can we do?
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@ Miguel Angel Montes
. ¿Can you try a definition of infinity that does not concern sets?
Well, infinity is a property of sets. Better said, cardinality is a property of sets, and infinity is a property shared by some cardinalities... Yes, it is hard to define infinity without using sets, but why should we do this? 
About analytic and mechanical notions concerning infinity, like "infinitely small movement", and so on: This is a quite different question from the first one, about potential and actual infinity. Here there is much more structure involved. Archimede's axiom tells that for every real number, there is a natural number that is bigger than it. Happilly, there are ordered fields which are non-archimedean. There you have field elements r which are bigger than any natural number. Than 1 / r will be positive but smaller than all 1/n , so this is an "infinitely small quantity" (like in non-standard analysis). As you see, infinitely small elements are a notion which cannot be befined without additional structures (ordered fields, order compatible with field structure, negation of Archimede's axiom, and so on) and not only a discussion about cardinalities, like the other point.
However, I agree with your arguments concerning psychology and philosophy, or the way children can learn about infinity. Fine psychological mechanisms maybe continue to make the difference between actual and potential infinity even by professional mathematicians. The most intense use of potential infinity appears, as I believe, when cosmologists try to think about an infinite universe and when analytical mechanicians try to think about an infinitely small movement...
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In literature, for real vectors, the relationship between higher order cumulants and moments can be found. 
How does this relationship look for complex vectors?
For example, how does E[y*1 y*2 y*3 y4 y5 y6] relate to all the higher order cumulants? Is there a general formula for this?
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There's no difference, it's just that you have to view X_i and X*_i as independent variables.
If you denote moments by M_ij.. = E[Xi Xj...] etc, and cumulants by C_ij..., then the general formula is, in words, that E_ij... is the sum over all partitionings of ij... of the corresponding product of cumulants. Thus, e.g.,
    M_i = C_i,
    M_ij = C_ij + C_i C_j,
    M_ijk = C_ijk + C_ij C_k + C_ik C_j + C_jk C_i + C_i C_j C_k.
These relation can be inverted to yield cumulants as a similar but slightly more complicated sums of products of moments, thus.e.g.
    C_i = M_i,
    C_ij = M_ij - M_i M_j,
    C_ijk = M_ijk - M_ij M_k - M_ik M_j - M_jk M_i + 2 M_i M_j M_k.
A nice way of summarizing this is in terms of generating functions - GF. For a single variable, if F(a) is the moment GF,
    F(a) = E[exp(a X)] = sum_n a^n E[X^n] / n!
then G(a) = log F(a) is the corresponding GF for cumulants. This has a straightforward generalization to several variables.
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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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(A) Let X\neq\emptyset and \tau\in P(x), where P(X) is the power set of X. Then if we have that:
i) X,\emptyset\in\tau
ii) i\in I, A_{i}\in\tau \Rightarrow \cup_{i\in I}{A_{i}} \in \Tau
iii) j\in J, J finite, A_{j}\in \tau \Rightarrow \cap_{j\in J}{A_{j}}\in \tau
We say that the collection \tau is a Topology in X and that (X,\tau) is a topological space.
(B) The minimum requirement for every algebraic structure is closure under the defined binary operation.
What is the essential difference between Algebra & Topology?
Both of them are guided by the concept of closure.
So, why have we defined two branches that are almost of the same philosophy?
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What about motivating intuitions? Topology was developed basically to deal with intuitions about "space," "connectivity, "continuity," notions of "near" and "far," etc. Algebra came about in order to deal with notions of "finitary manipulation," especially in connection with equalities. The historical developments happened quite organically, without any particular "guiding hand," so the boundaries are fuzzy indeed...
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In the attached paper it is a rather speculative position. I would like to have specialist opinion of the situations of today's position.
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Thank you Marika! Great post! I study it carefully.
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How can we introduce the idea of infinity to students? Its properties, relationship with zero etc.?
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Two mirrors - one looks to another.
In them there is no reflection, but infinity.
What does this mean? What is it?
How to explain? - Mystery of the mind.
Perhaps there lies the road to infinity?
Can’t see through the glass end of the road.
Unfortunately, in every science concept of infinity is different.
What kind of science is it?
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If we have a first order language for Real Numbers (this is exactly the language of rings), the using Dedekind cuts we can form a formula defining a real number. This is a syntactical determination of a real number. Some mathematicians stop here! They do not like models and interpretations. But if we try to find models, and interpret the formula, then the are infinitely many non-isomorphic models!
First let us take Boolean-valued models V^(B). Takeuti in “Two Applications of Logic to Mathematics” have prove that in a Boolean-valued model V^(B), if B is a complete Boolean Algebra, of projections in a Hilbert space, then real numbers are just self adjoint operators and if B is a probability algebra then the reals are “random variables”. In addition we have nonstandard models of reals that contain infinitesimals, and finally the are also tops theoretic models of real numbers.
Thus the real question is: How we can comprehend this infinitude of models of real numbers? We should note that completeness axiom is a second order axiom and thus if we want to stay in first order logic, we have to accept that there are infinitely many non-isomorphic models.
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There is never going to be a truly satisfactory answer to this question. Even the set N={0,1,2,...} of natural numbers is not categorical, as an ordered set, in the sense of first-order logic. Funnily enough, though, the (apparently more complicated) set Q of rational numbers is so categorical. One of the objections I have heard about nonstandard analysis is that it has no "standard" model; i.e., no standard construction--Dedekind-like cuts or otherwise--of a nonstandard real line. I don't find this objection very convincing, though: none of these things has a separate existence outside of a certain Platonic idealization.
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Are Automated theorem provers really useful to help students to know and understand how to proof? Is there any research about it?
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Yes, you're right. That's pretty hard without any operator terms available. I need 1 month just to understand how to proof classical logic properties like p -> q <-> ~q -> ~p using Coq.
Especially for Coq (I don't know about another ATP), sometime it's not take a long time to prove, but not simple as doing proof on paper. That's right, maybe ATP is not suitable for teaching (for high school student, it's impossible!) proof.
"I really believe a proof assistant would be better for you."
But, Coq is proof assistant too :o
I find another software called DC Proof 2.0, the proof-writing software not an ATP. It might be useful for teaching proof to students.
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Dear Colleagues, I am working on a new theory named Prelogic or Crealogic that uses the idea of sets and their creation for explaining the mechanisms that assemble intelligence and logic.
More interestingly, using Prelogic as the possible foundation of Set Theory, the Russell Paradox is no longer a contradiction or inconsistency of math and logic, but rather a clear illustration of the set forming dependency of an intellect.
Additionally, this so called Prelogic, can explain the reason why neurons connect and communicate the way they do to create intelligence.
Noteworthy, we have assembled a computational prototype of the above, that also uses sets for representing Language with mathematical-like proficiency, and soon, we hope to begin assembling the representations for Images.
For example, in language, Prelogic claims why is it that when we say "the red car and the yellow submarine," the car is red and not yellow, while the submarine becomes conceptually yellow but not red. What happens is that the meaningless entity representing "car" forms a SET with the meaningless entity representing "red." A similar thing happens to yellow with submarine, and therefore the reason why the submarine is not red nor the car is yellow. In this fashion Grammar becomes nothing more than binding rules (reason why grammatical elements are not freely exchanged) which only purpose is to acknowledge complex concepts (which concepts/words from SETS with other concepts/words)
So in the case of biological intelligence, neurons (meaningless cells representing concepts) form SETS which are identifiable only when communications travel along their connections (dendrites). Noteworthy, neurons can not draw areas around them like in Set Theory to create sets. The only way they can do this is by connecting with each other and identifying this connections by communicating.
Thanks
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Hello Frank, you might want to check out some basic foundational work in set theory, and logic, perhaps even formal languages. Getting some of the basic naming conventions down will help you in your research. For example, the difference between a class and a set was formalized in NBG set theory, and properties of sets, are the class of such set in that system, so for example, the set of cars that has a class of red, which is the property red applied to cars, and it seem to me this is very close to what you mention. Russell's paradox has been dealt with by both Zermelo and Neumann through two different axioms, and is not really a pressing issue in foundations. It is, on the other hand, a curious introduction to naive set theory. Also, one or more axioms which produce other deductive truths from those axioms, is not usually referred to as another axiom, instead, it is often called a theorem, lemma or a proposition, depending on the circumstance.
By "imagination", it seems to me you are trying to formalize the conjecture, which is noble and necessary research. For this, I suggest you look into the research on learning algorithms, which, through the analysis of data, conjectures about the properties of such data can be made, and the algorithm can make a decision based on this conjecture. It is how we get self-driving cars.
In terms of Godel's incompleteness theorems, The simplest way to communicate what he did is that he proved that there are true statements that can be formed in a formal language which can not be proven as true in that language, he also proved that all complete systems must contain a contradiction or it can't be complete, and thus, any formal system that is consistent, must also be incomplete.
At a glance, based on the little I have seen here, it seems to me that you have created a string generator where the strings themselves generate based on the consistency of previous strings. What you have here, is not necessarily truth in the language of English necessarily, but you have included, on a more structural level, truth in how strings may be formed, which just happen to coincide with statements in English that may or may not be true to an observer. Such a system, if this is the case, is still under the rules of Godel's incompleteness, but you need to be clear about the system from which you are making your research claims. For example, in your program that you seemed to have worked out, "Mary is a girl" is a theorem in the program, despite the fact whether Mary is actually a girl or not to an English speaker/observer. In this sense, you should want to separate the formalism of your program which is not English that generates strings from a subset of English, which as a program, has a consistency and is subject to incompleteness, from all statements which can be made in English, which is a language that is complete (in the sense it contains at least all of predicate logic) and inconsistent.
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Dedekind cuts establish the set of real numbers. But like the well ordering theorem, the comparison of two arbitrary real numbers and linear ordering or law of trichotomy defies intuition. Of course rational numbers can be compared. Mostly epsilons can be replaced by 1/2^n etc.It is interesting to note that Dieuodenne's treatise on Analysis always uses rational numbers in the above form rather than using arbitrary Epsllons. Actually real numbers only give a universal set.
We need a few irrational numbers like Pi, e, Gamma and square roots or n'th roots of natural numbers. So my question has a lot practical relevance . The foundations of analysis depends upon set of real numbers but not the actual transcendental real numbers save those specified in earlier sentence. Also one can use Continued fractions to express Pi , sq root of 2 etc. Will some researcher throw light on this? It is interesting to note that that Dieuodenne actually uses only rational numbers in his proofs? What was his philosophy?
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I beleive these notions are important only when they shed light on real numbers . intuiitve understanding which i remarked sheds enough light constructive real analysis a rg8 subject . y remarks about aha Banach theorem are very perfect reflected in many books . I request respected Prof Nerode to glance hroughprefaeof the classic foundations of modern analysis by J.Dieupdene. i beleive i alied computer science , compiler construction, or logic progrraming,, Analysis , but personally my tests do not incline to model theory or recursie functio theory as i feel their purpose is wel served intuitively . neither these theories are adding to applicatons of computer science (real world) or enrichment of Mathematics.
I Know turing machine , post machine URM are all equivalent but i feel one should list axioms or axiom schema for modelling computation rather than machines as hey fit more nicely into mathematical framework. this remark can be more appreciated by a mathematician than a computer scientist. my professor of computer science dr. S. K. Mehata when listening to my views remarked that ow can u model computation without a machine . Now that is a typical engineer or computer scientist mental framework. a mathematician would love it modeled axiomatically!.