Following

There are two major differences between the old and new infinite set theories (The Second Generation Of infinite Set Theory) .

It is well known that the founding of present classical infinite set theory is a brilliant achievement in our science history. Cantor leaded us human into a new field of “quantitative cognitions to infinite related mathematical things”. “’Infinitesimal, infinite greats and infinite many’ related mathematical things” should be qualitative-quantitative studied and cognized ------- infinite set theory is needed and there must be infinite set theory in human science.

The emergence of new infinite system determines the production of "new infinite set theory"--------The infinite set theory based on the classical infinite system is called "classical infinite set theory " while the infinite set theory based on the new infinite system is called "new infinite set theory (the second generation of infinite set theory)".

Different basic theories decide the some differences between new and old infinite set theories.

1, Different infinite ideas

In classical infinite set theory, “infinite” is composed by both contents of “potential infinite, actual infinite”. So, any infinite related “mathematical things” in present classical set theory basing on classical infinite idea can not run away from the fetters of “potential infinite, actual infinite” [13-22]. Although it is generally accepted that “actual infinite” is a kind of “tangible infinite”, no one in the world so far knows exactly how on earth “actual infinite” be “tangible” theoretically and operationally, and no scientific and clear definition has ever been given for “potential infinite-- actual infinite” since they came to human science and mathematics. The inevitable confusion of “potential infinite, actual infinite” and the absence of the whole “actual infinite theory” in the “potential infinite, actual infinite” based present classical set theory inevitably result in following fatal defect: it is impossible to know clearly what on earth the exact “infinite mathematical things” being quantitative cognized in set theory are and how to be treated (such as “are they potential infinite mathematical things or actual infinite mathematical things?”, “whether “potential infinite mathematical things or actual infinite mathematical things” should be treated differently?”, “if they should be treated differently, how to operate?”, “why?”, …). So, there are no scientific basis for many quantitative cognizing activities to “infinite mathematical things” in present classical set theory------- just do as one wishes to produce many inevitable errors and paradoxes.

In new infinite set theory, “infinite” is composed by both contents of "abstract infinite law and the carriers of abstract infinite law” (called “infinite” and “infinite carrier” for short following) which can be scientifically and clearly defined without any confusion [7-22]. The “infinite carrier” in new infinite idea is a kind of “visible and tangible science thing” and it is possible for us to construct a complete and systematic “theory of infinite mathematical carriers”------the infinite related new number forms (number carriers), new number spectrum, new set forms (set carriers), new set spectrum as well as the treating theories for those “infinite mathematical carriers”. This new “theory of infinite mathematical carriers” enable us to know theoretically and operationally what on earth many “infinite mathematical things (infinite carriers)” being cognized are, how they are “visible and tangible” and how to carry on the scientific operations of quantitative cognizing to them. So, it is impossible to have any theoretical and operational confusions of “potential infinite-- actual infinite” in new infinite theory system and, the “‘infinite paradox’ symptom complex” produced by the “confusions of potential infinite-- actual infinite” in present classical set theory is to be naturally eliminated.

2 Different quantitative cognizing operations and results to "infinite related mathematical things"

In old infinite set theory: nothing is able to run away from the constraining of “potential infinite--actual infinite” and all of its “mathematical things" can only be existing within following three "potential infinite--actual infinite" related contents: (1) "potential infinite mathematical things (such as potential infinite numbers, potential infinite sets, potential infinite elements of infinite set)" ; or (2) " actual infinite mathematical things (such as actual infinite numbers, actual infinite sets, actual infinite elements of infinite set)"; or (3) “the mixture mathematical things of potential infinite and actual infinite" which can be either "potential infinite” or “actual infinite" as one wishes at any time. All these “infinite related mathematical things" in present classical set theory have nothing to do with present classical number system and set system (there are not any “potential infinite--actual infinite” related number forms and set forms in present classical number system and set system), nothing to do with the specific quantitative properties of "infinite mathematical things" (What quantitative properties do potential infinite numbers, potential infinite sets, potential infinite elements of infinite set have? What quantitative properties do actual infinite numbers, actual infinite set, actual infinite elements of infinite set have? Can these infinite mathematical things be numbers or sets? ...). Such fundamental defects have been making us unable to understand scientifically and clearly the quantitative properties, meanings and forms of infinite mathematical things in present infinite related classical mathematics system.

So, theoretically or operationally, it is impossible for us to establish the scientific and systematic quantitative cognizing and operating theories on “infinite related mathematical things (such as quantitative cognizing and comparing theories for those "X----> 0 infinite related mathematical things and Y ---->📷infinite related mathematical things", the operating theories of one-to-one corresponding theory and limit theory, ...)” in present classical “potential infinite--actual infinite” based set theory. Such fatal fundamental defects naturally result in following two unscientific contents in present classical set theory:

（1）arbitrary infinite theory: only accepting infinities but denying infinitesimals; “infinite sets, elements of infinite sets, numbers of elements in infinite sets” can be taken as "potential infinite” or “actual infinite" or the mixture of both; ...);

（2）arbitrary operations and formal languages: any “infinite series” can be cut into pieces to produce super infinity, super super infinity, super super super infinity,... with different grades; never knowing how and when to use the formal languages and operations of “let be 0, take the limit, take the standard number” for those “X----> 0 infinitesimal elements” in [0,1] Real Number Set; an exactly same pipeline operation and formal language can be used to produce totally different products------- the “one-to-one corresponding operation and formal language” can not only be used for the result of “Rational Number Set is equal to Natural Number Set” but also for the result of “Rational Number Set is bigger than Natural Number Set”, “the pipeline operation and formal language of looking for something belongs to an infinite set but is impossible to be found inside this infinite set (the confusing mechanism of ‘potential infinite--actual infinite’)” can not only be taken as “fatal fundamental defects” in present classical set theory resulting in the production of Russell’s Paradox, but also as “important fundamental theory” in present classical set theory resulting in the production of modern “Power Set Theorem” and “Uncountability of Real Number Set Theorem” , ...

In new infinite set theory: the “new infinite carriers’ theory” closely relating to “new number forms, new number system (new number spectrum), new set forms, new set system (new set spectrum)” is an important basic theory for us in our quantitative cognizing on “infinite related mathematical things”. All the “infinite related mathematical things” being cognized are “mathematical carriers of abstract infinite law” with tangible quantitative properties and meaning, expressing the existence of abstract infinite law-------they are “mathematical things” with unequivocal quantitative properties and manifestation modes clearly defined by new number spectrum and set spectrum (such as inter-numbers and inter-sets). So, different from the inevitable confusion situations of “potential infinite” and “actual infinite” in old infinite set theory, it is impossible in new infinite set theory to have the confusion situations of “abstract infinite law” and “carriers of abstract infinite law”. But, the most important thing is, it never allows taking “law” as “carriers” for quantitative cognition like the old infinite set does. Being one of the “science carrier theories”, the newly developed “theory of infinite related mathematical carriers” as well as its relating “carriers analyzing and treating theories (such as ‘new limit theory’ and ‘new one-to-one corresponding theory’ relating to ‘infinite related mathematical carriers’)” fills in a fundamental blank for the “quantitative cognizing theory for infinite related mathematical things” in present classical mathematical analysis and set theory and open up a new quantitative cognizing field of “infinite related mathematical things”. The new basic theory ensures the Second Generation of Set Theory be free from any unscientific influences of "potential infinite--actual infinite" and eradicates above unscientific “paradoxes producing factors” as well as their production of “infinite paradox symptom complex”. Before practical quantitative cognizing operations in new set theory, those cognized “mathematical things” must be truly "mathematical analyzed" according to the newly developed “theory of infinite related mathematical carriers” to know what quantitative natures, meanings and manifestations they have so as to perform the targeted quantitative operations to them.

So, such targeted operating mechanism. method, process and result decided by the newly developed “theory of infinite related mathematical carriers” as well as its relating “carriers analyzing and treating theories” in new infinite set theory are surely nonequivalent in many situations to those operating mechanism. method, process and result decided by “potential infinite--actual infinite” in present classical set theory-------it is impossible for the Second Generation of Set Theory to have (to produce, nourish and protect) those suspended paradox families relating to the quantitative cognitions of "infinitesimals and infinities" as the first Generation of Set Theory has.

Except those unscientific contents caused by the confusion of “potential infinite--actual infinite”, the Second Generation of Set Theory retains most of the things in the first Generation of Set Theory.

Nice question.

Thank you dear Dr. Peter Breuer,

How do you answer following two infinite related problems?

1, Are the concepts of “potential infinite--actual infinite” important in present classical infinite related science system? If yes, what roles they play; if not, why they have been existing in our science?

2, Can we really produce infinite items (numbers) each bigger than 1/2 or 100 or 1000000000000000 or 1000000000000000000000000000000 or… from the Harmonic Series 1, 1/2, 1/3, ... by the operation of "bracketing” with limit theory and turn the Un--->0 Harmonic Series into a “Vn ---> any positive constants” infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000)？

Yours,

Geng Ouyang

Thank you dear Dr. Peter Breuer,

How do you answer following reedited infinite related questions?

1, Are the concepts of “potential infinite--actual infinite” important in mathematics (science)? If yes, what roles they play; if not, why they have been existing in mathematics (science)?

2, how many items in Harmonic Series you operate scientifically for the first 100000000000000000000000000000, how many for the second 100000000000000000000000000000, how many for the third 100000000000000000000000000000, how many for the fourth 100000000000000000000000000000, how many for the fifth 100000000000000000000000000000, ... and turn the Un--->0 Harmonic Series into a Vn ---> any positive constants’ infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000)?

Yours,

Geng Ouyang

Dear Geng. What you meant under classical set theory?

ZFC, GB,NF, IST, Categorical set theory, etc.???

In contemporary mathematic usually we deal with ZFC+Con(ZFC)!

see also

https://arxiv.org/pdf/1212.6543.pdf

>Are the concepts of “potential infinite--actual infinite” important in mathematics (science)?

No at all. That is only philosophical question.

For example:

Harmonic Series

1+1/2 +1/3 +....+1/n +....

diverges potentially and actually but only divergence

important here in mathematical context.

http://vixra.org/abs/1605.0082

There are no any fundamental defects in classical set theory except Big one:

~Con(ZFC)!

Article There is no standard model of ZFC

>That is gibberish. I have no idea what it means. What is a "harmonic series"? Where have you got this nonsense phrase from? Why do you think it means something, and what do you think it means, and why aren't you saying? And what does "operate scientifically" mean?

Dear Peter Breuer, harmonic series is:

1+1/2 +1/3 +....+1/n +....

http://mathworld.wolfram.com/HarmonicSeries.html

Thank you dear Dr. Jaykov Foukzon and Dr. Peter Breuer,

1, for “classical infinite set theory”, I mean any infinite set theory with the foundation of “potential infinite--actual infinite” ideas.

Our infinite related science history has proved that the concept of infinite is composed by both “potential infinite” and “actual infinite”. On the one hand, no one is able to deny the qualitative differences and the important roles “potential infinite--actual infinite” play in the foundation of present classical infinite theory system; on the other hand, no one is able to deny that the present classical set theory is basing on “potential infinite--actual infinite” concepts as well as its related whole present classical infinite theory system. The fact is: any areas in present classical infinite related science system (of cause including present classical mathematical analysis and set theory) can not run away from the constraining of “potential infinite--actual infinite” concepts-------all the contents in present classical mathematical analysis and set theory can only be existing in the forms of “potential infinite mathematical things” and “actual infinite mathematical things”.

Of cause, many mathematicians try to deny and run away from the concepts of “potential infinite” and “actual infinite” just because “in so doing, people can treat ‘infinite related mathematical things’ very very freely”.

2, dear Dr. Peter Breuer: (1) “completed infinite” and "actual infinite" are the same thing in present science (including mathematics); (2) I am very sure that you know what Harmonic Series is and how it is proved to be divergence very well. My question is very clear: how many items in Harmonic Series you operate for the first 100000000000000000000000000000, how many for the second 100000000000000000000000000000, how many for the third 100000000000000000000000000000, how many for the fourth 100000000000000000000000000000, how many for the fifth 100000000000000000000000000000, ... and turn the Un--->0 Harmonic Series into a Vn ---> any positive constants’ infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000); (3) our mathematical history tells: only scientific things "exists" in mathematics and those unscientific things do not (will not) "exists" in mathematics.

Yours,

Geng Ouyang

.Edvard Nelson:Warning Signs of a Possible Collapse of Contemporary Mathematics.

https://web.math.princeton.edu/~nelson/papers/warn.pdf

Thank you dear Dr. Jaykov Foukzon.

Yours,

Geng Ouyang

Thank you dear Dr. Peter Breuer, just take it easy--------we are discussing a question “Looking for the way to solve the fundamental defects in classical set theory?” here.

I wish you can go and check how the Harmonic Series (it doesn’t matter even if you think it is “an old, unused, name for the sum of the reciprocals of the natural numbers”) is proved to be divergence, than you are able to understand my question: how many items in Harmonic Series you use for the first 100000000000000000000000000000, how many for the second 100000000000000000000000000000, how many for the third 100000000000000000000000000000, how many for the fourth 100000000000000000000000000000, how many for the fifth 100000000000000000000000000000, ... and turn the Un--->0 Harmonic Series into a Vn ---> any positive constants’ infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000).

Yours,

Geng Ouyang

"Mathematics that is manipulation of symbols according to agreed upon formal rules and nothing more"

David Hilbert.

>Interesting paper ...

Yes...But Tao found some mistakes in this Nelson's paper!

There exists an discussion on FOM, etc. about this paper.

https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html

Newerzeless PA is inconsistent as Edvard has told in this paper

http://lesswrong.com/lw/7u2/edward_nelson_claims_proof_of_inconsistency_in/

At 2013 I have posted on FOM another different proof about

~Con(ZFC+E(omega model of ZFC))

see discussion

https://cs.nyu.edu/pipermail/fom/2013-February/016995.html

>Generally, adding a model of ZFC of whatever kind to ZFC would be consistent. That is, if it were not consistent (as you state), then, assuming ZFC is consistent, the inconsistency must be in the axioms that the model models. If that is just ZFC's axioms, then that is a contradiction, as we assumed ZFC is consistent in reasoning thus far.

Yes of course.

Interestingly, superexponentiation shows up very often on lesswrong. Maybe that's a good answer to XiXiDu's question. If Nelson is right, then people should stop using 3^^^3 in their thought experiments.

Nelson was right, and therefore even PA revision needed.

Peter ：

1，Zeno’s Achilles and Tortoise Paradox is only dealing with primary school mathematics, even more “simple and easy” to you. I guess you dare not to laugh at Zeno, don’t you?

2, It is good that you have learned what Harmonic Series (“an old, unused, name for the sum of the reciprocals of the natural numbers”) is and how it is proved to be divergence, but I never say the Harmonic Series is convergent-------it is not that simple as you imagined.

3, Are you still so sure that you are able to turn the Un--->0 Harmonic Series into a Vn ---> any positive constants’ infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000)?

Of cause， “I can't be bothered” is always an excellent excuse to run away from challenges.

1, It seems you are really an expert in the infinite related mathematical areas with so many unique and great ideas on Zeno’s Achilles and Tortoise Paradox.

OK, as you are used to “I have no idea what that means” on many infinite related mathematical things, I now make my question in a very very simple way:

Can you produce infinite 100000000000000000000000000000 from the Un--->0 Harmonic Series?

2, It seems you are really a good teacher caring so much for students and teaching affairs, me too and I would like to join any of your topics about students and teaching affairs. But we are here just discussing the infinite related mathematical things on the topic of “Looking for the way to solve the fundamental defects in classical set theory”, aren’t we.

to Peter Breuer

https://www.researchgate.net/post/Was_Edvard_Nelson_wrong

Dear all. Remind Euler deal with infinite summ

s=1+1/2 +1/3 +....+1/n +....

under assumption that s is a well defined infinite number

http://eulerarchive.maa.org/hedi/HEDI-2005-02.pdf

Thank you Peter.

1, I have mentioned many times the Un--->0 Harmonic Series in this line, remember?

2, I am very surprised that you still don’t know what “1+1/2 +1/3 +....+1/n +....” is, how it is proved to be divergent and what "100000000000000000000000000000" is in this line so far.

3, Of cause, “beating around the bush” is also an excellent excuse to run away from challenges.

Now I understand at last the way you do here-------copying down chapters from the books to define each word before writing. Well, it might be a good habit for you and you can do whatever you like of cause.

One suggestion: “I have no idea what that means” not became your pet phrase, and “beating around the bush” not became your habit when you meet challenges.

How do you DECIDE that I only know one proof?

It seemed you are used to be so strictness on each word but in fact you insult others again and again at your own will so freely. This is unfair, isn’t it?

“Beating around the bush” again!

Well, although I am not always pretend to be a great linguistician like you, let me tell you what mathematical meaning is for the sentence of “I am very surprised that you still don’t know what 1+1/2 +1/3 +....+1/n +.... is and how it is proved to be divergent”, this sentence has following two meanings:

1, you don’t know what 1+1/2 +1/3 +....+1/n +.... is;

2, you don’t know how 1+1/2 +1/3 +....+1/n +.... is proved to be divergent”.

This sentence has nothing to do with “how many ways of proving”, it doesn’t tell you at all there is only one way to prove 1+1/2 +1/3 +....+1/n +.... is divergent.

I hope you don’t use your pet phrase this time and say “I have no idea what that means” again.

Thank you dear Dr. Peter Breuer for your praised “your sentence structure and grammar has improved”.

I am not sure where you learned your English, but in the sentence of "you don’t know how 1+1/2 +1/3 +....+1/n +.... is proved to be divergent”, that very "is proved" means the subject is singular, that very "is proved" really has nothing to do with “how many ways of proving” at all, it doesn’t tell you at all there is only one way to prove1+1/2 +1/3 +....+1/n +.... is divergent. I can’t help if you still use your pet phrase and say “I have no idea what that means”.

I am very sorry that in the topic of “Looking for the way to solve the fundamental defects in classical set theory” I accompany you “Beating around the bush” and spend so much time to discuss how to become a "great linguist".

For your question “WHAT proof is it that you think I do not know?”. To tell the truth, I think you do not really understand many “infinite related mathematical things” (including their relating proofs). But this is not strange because I know very well myself that I only know little in your field and I also know you are a real expert there.

Is it true “Difference in profession makes one feel worlds apart”?

Yours,

Geng Ouyang

1, It seems you are still so sure that you know much more than me about “infinite related mathematical things” (including their relating proofs). To tell the truth, I think you can only copy something form books those stuff of “confusing potential infinite--actual infinite” since Zeno’s time.

2, I never say I know much than you in you field, remember? Do you know a saying: “Difference in profession makes one feel worlds apart”? It is now my turn to tell you “Beware that you may be experiencing one of the rarer forms of Dunning-Krueger syndrome”.

Thank you dear Dr. Peter Breuer,

For “State something I have copied from books those stuff of confusing potential infinite--actual infinite since Zeno’s time”, those you have presented here about “infinite” are from books----- those stuff of confusing potential infinite--actual infinite since Zeno’s time (you even didn’t copy well enough as you deny many well acknowledged paradoxes). But don’t worry about that I did exactly the same 40 years ago.

You are modesty to say “I don't have a field". Thank you.

The way to solve first three conundrums presented in On the Quantitative Cognitions to Infinite Things III:

1. Does the definition of each “infinite set” closely relate to the elements’ “special nature, special existing condition, special manifestation and special relationship among each other” ---------the very characteristics of the elements inside the very “infinite set”?

2. If it does not at all and the elements in all different infinite sets are the same a heap of “infinite abstract stuff without any differences and relationship”, then how can we define and distinguish “different infinite set” and how can we believe that there may be quantity differences between “different infinite sets”?

3. If it does and the elements in all different infinite sets are “the concrete carriers of ‘abstract infinite concept’ with differences and relationship” (unique characteristics), then how will these unique characteristics decide the quantity differences between “different infinite sets”?

These three conundrums have disclosed from another angle the exactly same “quantitative cognizing defects of infinite related mathematical things” in analysis also caused by the confusion of “potential infinite--actual infinite”, mainly involving the cognizing defects to two important contents of “infinite set” and the “elements of infinite set” as well as their relationship (we have discussed those defects in above topics). Such fundamental defects have been unable us to carry on the scientific and systematic quantitative cognitions to “infinite sets and the elements in infinite sets” so as to produce many operational erros and paradoxes in present classical set theory such as: “constructing a set with elements belonging to it but impossibly to be found in it” for all the family members of Russell’s Paradox, “taking infinite set elements as a heap of abstract infinite points without special nature, special existing condition, special manifestation and special relationship among each other” for the proof of “cardinality inequivalence proof of motherset and their sub sets” (such as Rational Number Set’s countability) and the completely opposite operation of “taking infinite set elements as tangible mathematical carriers with special nature, special existing condition, special manifestation and special relationship among each other” for the proofs of “cardinality inequivalence proof of motherset and their sub sets” (such as Real Number Set’s uncountability as well as the Power Set Theorm), ...

The new infinite idea of “abstract infinite law and carriers of abstract infinite law” is the foundation of the new infinite theory system. With the abolish of the mistaken and unscientific “potential infinite and actual infinite” concepts, the new infinite theory system based SECOND GENERATION OF INFINITE SET THEORY will never be entangled, influenced and constrainted by the not-knowing-what “potential infinite--actual infinite”. So, all the “potential infinite--actual infinite” related conundrums and paradoxes in the FIRST GENERATION OF INFINITE SET THEORY are to be solved thoroughly. The same as in the mathematical analysis, “infinite sets” and “infinite setss elements” can only (have to) exist in human mathematics as “infinite related mathematical carriers”-------the different characteristics of “mathematical carriers” in infinite set theory decide many differences among different infinite sets (including different numbers of their elements).

I think crispness is a defect for classical set theory.

> I think crispness is a defect for classical set theory.

There is only one defect of the classical set theory: that is abnormal assumption Con(ZFC), etc.

But, why is there such an abnormal assumption Con(ZFC)?

Why it should be correct?

If you think there is “one defect” in classical set theory, why it should not be correct?

>why it should not be correct?

Because obviously ~Con(ZFC).

To Dr. Jaykov Foukzon,

How and what is “obviously”?

to Peter Breuer

>it is not an "assumption", so much as widely believed to be true but

unprovable]

any belief is an assumption. for example belief in santa claus

mean that santa claus really exist!

Con(ZFC) meant there exst model M of ZFC.

See for example: Set Theory and the Continuum Hypothesis (Dove… (Paperback) by Paul J. Cohen. section 6 https://www.amazon.com/Theory-Continuum-Hypothesis-Dover-Mathematics/dp/0486469212

Such model is an sort of "santa claus" in which mathematichians widely believe.

Usually we dealing under assumption (postulate): there exst standard model Mst of ZFC.

However in set theory ZFC+∃Mst provable ~Con(ZFC+∃Mst )!

http://www.sciencedomain.org/abstract/22940

However such that possible: there exst some nonstandard model MNst of ZFC.

>How and what is “obviously”?

https://www.researchgate.net/deref/http%3A%2F%2Fwww.sciencedomain.org%2Fabstract%2F22940

>I believe that ZF is consistent, but I do not assume it.

You assume it by using postulate: there exst model M of ZFC.

>Con(ZFC) meant there exist model M of ZFC." No, it means that THERE IS NO PROOF OF FALSITY in ZFC.

"THERE IS NO PROOF OF FALSITY in ZFC" is equivalent to "there exist model M of ZFC" by Godel completeness theorem

>And I repeat, your claim that Con(ZFC) means displaying a model of ZFC is false.

No at all.

See for example: Set Theory and the Continuum Hypothesis (Dove… (Paperback) by Paul J. Cohen. section 6 https://www.amazon.com/Theory-Continuum-Hypothesis-Dover-Mathematics/dp/0486469212

To Dr. Jaykov Foukzon,

Can you find out what problem is and open up a new way to “make it true”?



>Please expand your claim '"THERE IS NO PROOF OF FALSITY in ZFC" is equivalent to "there exist model M of ZFC" by [the] Godel completeness theorem'.

THERE IS NO PROOF OF FALSITY in ZFC meant Con(ZFC).

But ZFC+Con(ZFC)----> there exist model M of ZFC by [the] Godel completeness theorem.

Of course the statement "there exist model M of ZFC" unprovable in ZFC.

>PROOF OF FALSITY

Notice that even for definition of formal first order theory countable set of standard objects is needed : standard naturals etc.

https://en.wikipedia.org/wiki/First-order_logic

To Geng Ouyang

Please expand your question

To Jaykov Foukzon

Is there anything to do with “potential infinite” and “actual infinite”? We human have been torture by “potential infinite” and “actual infinite” for thousands of years, we have to open up a new way to free us from this abysm.

Very informative conversation which i have ever seen.

Our infinite related science history has proved that the concept of infinite is composed by both “potential infinite” and “actual infinite” since it came to our science. On the one hand, no one is able to deny the qualitative differences and the important roles “potential infinite--actual infinite” play in the foundation of present classical infinite theory system; on the other hand, no one is able to deny that the present classical set theory is basing on “potential infinite and actual infinite” concepts as well as their related whole present classical infinite theory system. The fact is: any areas in present classical infinite related science system (of cause including present classical mathematical analysis and set theory) can not run away from the constraining of “potential infinite and actual infinite” concepts-------all the contents in present classical mathematical analysis and set theory can only be existing in the forms of “potential infinite mathematical things” and “actual infinite mathematical things”.

To Peter Breuer.

Godel completeness theorem asserts:

ZFC+Con(ZFC) there exists some model M of ZFC

That is a theorem but not a philosophical statement.

I confirm that in set theory ZFC+∃M_st provable ~Con(ZFC+∃M_st )!

Of course set theory ZFC+∃M_st essentially stronger then the ZFC.

Example : let k be an inassesible cardinal then

I confirm that in set theory ZFC+∃k provable ~Con(ZFC+∃k)!

Without a sound foundation of scientific infinite theory system, Non-Standard Analysis, Typ Theory and Model Theory can do nothing to help us solve those suspended infinite related paradoxes produced by the confusion of “potential infinite-actual infinite” in present mathematical analysis and set theory.

To Peter Breuer.

>is meant to stand for the "gate" symbol "|-"?

Yes of course.

>because the cardinal supplies such a model

Inaccessible cardinal supplies ONLY such a STANDARD model!

Unexistence such standard model does not imply ~Con(ZFC)!

In this paper

http://www.sciencedomain.org/abstract/22940

in particular I proved that: if ZFC is consistent then only very exotic nonstandard models of ZFC exist.

>Any model of ZFC in ZFC (weird!) would be "exotic" anyway

No at all. There exist non exotic standard models of ZFC and exotic non standard models of ZFC , for example countable models given by Skolem theorem.

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)