Set theory is consistent at least according to the semantic formulation of consistency.

I would like to quote the great mathematician F. Ramsey, when he was asked for this question:

"Suppose a contradiction were to be found in the axioms of set theory. Do you seriously believe that a bridge would fall down?"

Frank Ramsey (1903 - 1930)

In addition, take into account, that theories are based upon axiom sets; however, the underlying deriving procedures are not consequence of the corresponding axioms. They depend on the applied logic rules. Accordingly it is not always easy to know whether contradictions are a consequence of theory axioms or logical methods. Even, maybe, of construction procedures.

For instance, if a contradiction were found in number theory, I prefer to change logic instead of axioms of arithmetics.

In fact, fuzzy sets are contradiction-free because every contradiction can be avoided in the scope of multivalued logic, simply, assigning fractionary truth-values.

In any case, there are several set theories. I recommend to read this article by Zenkin.

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.rml/1203431978

http://www.ccas.ru/alexzen/papers/CANTOR-2003/Zenkin%20BSL-2.pdf

or this one

https://www.researchgate.net/publication/258316557_Yet_another_paradox_in_Set_Theory._Cantor's_Theorem_revisited

Article Yet another paradox in Set Theory. Cantor's Theorem revisited

I confirm that the question is more subtle. Peano arithmetic in your reasoning has no proof of consistency. But G. Gentzen found a way to prove its consistency. It is a proof in the same way what we are expecting for the case of set theory.

To Cesar Rodrigues:

Regarding your original questions, ``Is set theory consistent? '', I say, ``Yes, but I cannot prove it at this time.''

Regarding your contention that G\"{o}del and Cohen ``provided models'' of set theory, you are incorrect. What G\"{o}del provided in his proof of the relative consistency of the GCH is a ``model'' whose existence is contingent upon the consistency of set theory. The reason it's existence is not absolute is that G\"{o}del pre-supposed the existence of a model V of ZF in order to ``construct'' within that given model V a model of ZF+GCH, namely the ``constructible universe'', L\subseteq V. If set theory were inconsistent, then G\"{o}del's construction of a model of ZF+GCH would fail. This is the backbone of the notion of a ``relative consistency'' proof. Similarly, Cohen's proof of the relative consistency of ZF+not(CH) pre-supposes the existence of a model of ZF. Nicely, though, Cohen introduced a new proof mechanism into set theory and foundations, namely the notion of forcing.

Regarding your contention that Getzen ``proved'' the consistency of arithmetic, no. His proof is in infinitary logic, specifically and infintary logic that is undecidable itself. Basically, he allows infinitely long sentences or infinitely long proofs. Either way, one needs to pre-suppose that Peano arithmetic (or some undecidable frament thereof) is consistent, in order to ``write'' the proofs.

To Juan-Esteban Palomar Tarancon:

I say FIE! Consider the theory of groups, which is certainly consistent. Let Th(Groups) denote this theory. Similarly let Th(AbGroups) denote the theory of abelian groups, which adds to Th(Groups) the hypothesis that all models of the theory in question are commutative. Now suppose that an engineer things all groups of matrices commute and use that assumption to build a bridge. It is then certainly possible that at a given time t during the life of that bridge, the groups of matrices governing its response to the physical forces being applied to it are commuting groups, and it is possible that at some time s>t, there is a noncommutative group of matrices that governs its response because the physical forces being applied to it have changed. It is possible then that the bridge collapses at that time s, or shortly thereafter. This does not demonstrate the inconsistency of the theory Th(Groups), and it does not demonstrate the inconsistency of Th(AbGroups), for both are consistent. However, it does demonstrate the inconsistency of the theory in the head of that engineer which appended to the theory of sets the claim that all groups are abelian.

Now, suppose that Set Theory is inconsistent. Then we can PROVE that all groups are abelian, and if the hypothetical (and rather contrived...) scenario I described above pertains, namely, if at time t, the groups of matrices governing its response to the physical forces being applied to it are commuting groups, and at some time s>t, there is a noncommutative group of matrices that governs its response because the physical forces being applied to it have changed. It is possible then that the bridge collapses at that time s, or shortly thereafter. If the engineer believed the false claim that all groups are abelian because someone PROVED it to him using an inconsistent set theory, then he would rightly blame the set theory for causing him to build that bridge so that it collapsed.

So Ramsey, brilliant as he was, misleads in the quote you presented. For if we found Set Theory to be inconsistent, the KNOWLEDGE of such a discovery is not necessarily going to cause some bridge to collapse, but in fact, said knowledge may lead us to question WHICH bridges are safe to travel upon, because we need to KNOW that the engineers used CONSISTENT MATHEMATICS to construct bridges, in order to TRUST the methods of construction of such bridges, and it is entirely possible that if an engineer builds a bridge using a part of mathematics that is derived from an inconsistent set theory, then the principles of construction used are themselves inconsistent, or inconsistent with the state of the world in which the bridge is built. Thus, to Ramsey I also cry ``FOUL!''.

Regarding your contiention that the logic could be at fault for contradictions that may be found in number theory:

You are incorrect. The logic used for the version of number theory which is to be considered in such discussions is First Order Predicate Calculus (FOPC, for brevity). This logic system is sound and complete. Thus if in first order number theory, a contradiction is someday found, it cannot be due to the logic being used. There are only two possible reasons for such a contradiction: (a) human error in applying the rules of FOPC (in which the ``contradiction'' is really a blunder, and not a contradiction in number theory after all...) or (b) inconsistency of the first order Peano axioms. Fuzzy Logic does not help mend contradictions in First Order Number Theory. It may help, however, to USE or STUDY higher order number theory, by allowing more use of ``natural language'' to reach some conjectures faster than if one is hindered by first-order complexity issues.

Your contention that fuzzy logic eliminates all contradictions is faulty, for it depends significantly upon the structure of the lattice of truth values for your chosen fuzzy logic system. Specifically, if that lattice has a least element, and if the semantic mapping that assigns truth values to formulas in some given language, then rules for negation that yield the truth value equal to the least element for some sentence that is a consequence of an axiom set would cause a reasonable fuzzy logician to conclude that the theory at hand is in fact inconsistent, even in that chosen fuzzy sense.

To Peter Breuer:

You're more right, and although there are some subtleties to the notions of incompleteness, these are not they.

Please read these references:

1. Foundations of Set Theory, in the series Studies in Logic and the Foundations of Mathematics, by A.A. Fraenkel, Y. Bar-Hillel, A. Levy. (available from google books at the following link: http://books.google.com/books/about/Foundations_of_Set_Theory.html?id=ah2bwOwc06MC)

2. On Formally Undecidable Propositions of Principia Mathematica and Related Systems, by Kurt G\"{o}del. (available from google books at the following link: http://books.google.com/books?id=0R9oDmaqmNUC&printsec=frontcover&dq=undecidability&hl=en&sa=X&ei=rVV9Us6INO6GyQGg_YDoCw&ved=0CEEQ6AEwBTgK#v=onepage&q=undecidability&f=false)

3. Computability and Unsolvability, by Martin Davis. (available from google books at the following link: http://books.google.com/books?id=nbOqAAAAQBAJ&printsec=frontcover&dq=inauthor:%22Martin+Davis%22&hl=en&sa=X&ei=PVZ9Uu3IDKn4yAH574C4BA&ved=0CFEQ6AEwBA#v=onepage&q&f=false)

4. Metamathematics of Fuzzy Logic, by Petr Hájek. (available from google books at the following link: http://books.google.com/books?id=Eo-e8Pi-HmwC&pg=PA15&dq=fuzzy+logic&source=gbs_toc_r&cad=4#v=onepage&q=fuzzy%20logic&f=false)

5. Works that cite Mai Gehrke and her collaborators' work on fuzzy sets and such: https://www.google.com/search?tbm=bks&hl=en&q=mai+gehrke#hl=en&q=mai+gehrke+fuzzy&tbm=bks

6. Introduction to Mathematical Logic, Fourth Edition, by Elliott Mendelson. (available from google books: http://books.google.com/books?id=ZO1p4QGspoYC&dq=schoenfield+logic&source=gbs_navlinks_s)

7. Mathematical Logic, by Joseph R. Shoenfield. (available from OCLC WorldCat: http://www.worldcat.org/title/mathematical-logic/oclc/526343)

Um, there isn't really any connection between consistency and bridges (that is what Frank Ramsey was pointing out). Perhaps there is some disagreement about what consistency actually means, and whether its use in the question has a slightly different meaning. While you can demonstrate consistency with an appropriate model, you have to have somewhere to prove that observed properties of the model are provably universal in the model, which is part of where things go wrong that were identified by Godel (incompleteness, inconsistency)

There are lots of relative consistency theorems, and I don't think that mainstream Mathematics seriously doubts that Set Theory is consistent, but remember, we've been burned before (Russell's Paradox), so we should be careful.

For an interesting take on the power and danger of self-reference (remember, that is the essence of Russell's and Godel's separate arguments), look up Non-Well-Founded Set Theory (Peter Aczel), and the excellent description of it in the book The Liar: An Essay on Truth and Circularity (Jon Barwise, John.Etchemendy)

There is proposed by Godel himself another possible proof of the consistency of arithmetic using functionals, in the 50's of last century.

Dear all,

Every test for consistence is always a dogmatic doctrine. It is based upon the dogma that logical axioms are Universal Truths.

However, there are several possible logical systems. For instance, in fuzzy logic there is no room for consistence or non-consistence. If a predicate leads to its negation, P => ~P, then assigning the truth-value 1/2 to both P and ~P, everything works fine. Accordingly, under fuzzy logic, the only question can be whether the assigned truth-values are adequate.

Dichotomic logic can only be accepted provided that it is successful in our experience. Roughly speaking, logic is an empiric science; otherwise, what is the device to prove logic laws? Logic itself? Can logic prove its own laws?

Notice that the underlying algebraic structure of logic consists of boolean algebra. The theorems of boolean algebra are proved by logic. This is a cyclic argument. Logic laws are true because boolean algebra is true. Boolean algebra is true because logic laws are true. The only alternatives are: 1) invoking experience. 2) Invoking the authority of a guru, say Gödel. Between both alternatives, I prefer experience.

This is why, if a contradiction were found in arithmetic or algebra, I prefer to change the underlying logic instead of the addition table. Accordingly, the proper question must be: Is dichotomic logic the adequate device to handle set theory, or we had better assume a multivalued one?

What is the status if we restrict us to 'Inherediarily finite sets'?

These seem to be the proper framework for all Computer Science and Physics.

Sorry 'herediarily finite sets' is the name. Tanks Peter.

I completely agree with juan esteban palomar tarancon (do you have a short name we can use?) - when the math doesn't work, change the math (but be careful); when the logic doesn't work, change the logic (but be very careful)

the issue seems to me to be a conflict between how much we can say and how much we can prove - i think it is well-known that expressiveness and provability are direct opponents, and each problem solver needs to determine where on their conflict dimension it needs to be

start with a problem

choose the mathematical methods that may be appropriate

choose the logical system that may be useful

remember, the problem comes first (not the math or the logic)

@Peter

of course, any integer is representable as a hereditarily finite set. I think that this is what a software specification language means when it has the integers among the available types. To have a set of integers, which again can be used as an element of sets is something different. At least C++ does not allow such a usage of its integer type. A nice observation is that each hereditarily finite set has a normal form as a finite string of braces (of course, for each opening brace there has to be a closing one, but some more rules have to be obeyed). I would be interested in knowing what one knows about consistency or other foundational questions in this simplest conceivable set theoretical system.

Dear Christopher,

I know that my name is long. Spanish people have two names, the father's one followed by the mother's name. Spanish women preserve their names when get married. However, you can call me, simply, Juan.

Peter,

thank you for your instructive answer. The Spivey document is really interesting stuff. He does not stop with countable since he has the powerset P(S) of each set S and card(P(N)) > card(N). But he takes all the infinit stuff not serious (since obviously not implementable) and stresses the subset of 'finitary' objects and constructions. If he would confine everything to the finitary subset, he would probably loose nothing important. He would, however, be burdened with an inconveniency which he perhaps hates: The whole edifice would depend on a 'global variable' e.g. the maximum number of bits that is available for coding an element of N. Then N would be a finite set and P(N) would have more elements than elements are in N. I don't think that it is difficult to find a formalism to handle such cases elegantly. The most obvious, and probably perfect solution would be to define some quantities as having value 'nil'. Or one would agree to understand all N-valued results modulo the largest n. Then one could still give a value for card(P(N)).

Let me recall a part of my motivation for being interested in this stuff. All the Gödel-based arguments which educated mathematical laymen mantra-like raise against all mathematics based results (RG debates have plenty examples of that) would plainly be non-applicable to a rich, and for all natural sciences fully sufficient subset of mathematics. It is also a reaktion to my memories of times when I was too young to have built some math-related self-confidence, and was mentally terrified by the chance that mathematics could be fundamentally flawed (since, as Gödel told us, the contrary cannot be proved in a world of actual infinity).

I found a set theory developed by Church !, which is consistent by construction.

See the wikipedia.

The set theory I mentioned before is called Chuch´s Set Theory with a universal set. I will quote the abstract of a paper on Church's Set Theory, by Thomas Forster:

"A detailed and fairily elementary introduction is given to the techniques used by Church to prove the consitency of his set theory with a universal set by constructing models of it from models of ZF. The construction is explained and some general facts about it proved."

Dear all,

Consistence lies in the scope of metamathematics that is partly maths and partly philosophy. Recall that philosophy lacks in the apodicticity of maths. This is why to assume different theories is a matter of personal preferences. In addition, there is more than one logical system. There are dichotomic logic, fuzzy logic, multivalued logic, and linear logic....etc.

There is no room for the concept of consistence in multivalued logic.

In multivalued logic when a predicate P is equivalent to its negation ¬P, assigning the truth-value 1/2 to each of them the contradiction vanishes.

Peter,

one remark more on heriditarily finite sets, only to make sure that in case your thoughts would come back to the topic anytime, you don't have a wrong picture of my propoasal.

The restriction to some finite bit number was only proposed as a device to be used in an implementation of the system. As a mathematical system the number of decimal (or binary) digits is finite for each integer, but there is no limit for this number. So, for instance, the product of any two integers is again an integer of the system. I'm sure (since I did it on my first basic computer) that one can define the arithmetics of decimally represented integers by algorithms for which it is evident that they 'halt', so that no 'proof by induction' is needed. Let me call this Riese arithmetics (after Adam Riese who made digidwise algorithmic arithmetics popular in Germany in ~1500). All Gödel incompleteness results refer to systems which 'contain Peano arithmetics' which require the principle of induction (or some kind of recursion in other formulations). So I'm quite sure that it is only a question of diligence to proof the consistency of Riese arithmetics. Do you know a result on that? Do you understand why logicians prefer Peano arithmetics over Riese arithmetics, when the arithmetical functions give the same results in both cases, and Riese would make at least arithmetics immune against Gödel arguments?

To Christopher Landauer:

You wrote: ``Um, there isn't really any connection between consistency and bridges (that is what Frank Ramsey was pointing out). ''

I seriously disagree. Consider the following reference:

General Theory of Bridge Construction, by Herman Haupt, A.M., Civil Engineer, published in 1865 by D. Appleton and Co., New York

From the Preface, namely page 8, 12 lines from the bottom:

``It was a long time before the fortunate discovery was made that the intrados might be of any form most pleasing to the eye, and that the conditions of equilibrium could, in general, be satisfied by making the joints of the voussoirs perpendicular to the ine of direction of the pressures; a fact so simple and obvious, that there is reason for surprise that it was not suggested to the first mind in which originated the idea of an arch of equilibrium.''

Here, Haupt is discussing the CONSISTENCY of a certain geometric model, from a PRACTICAL standpoint of aesthetic bridge construction. It is a shame that the sentence prior to the above observation of a mathematical fact happened to denigrate the mathematicians of the time vis-a-vis: ``Mathematicians have apparently exhausted their ingenuity in devising modes of distributing the weights so as to produce an equilibrated curve of suitable form for the intrados of an arch; but many of their speculations are far more curious than useful, whilst practical men have been disposed to reject the principle of equilibration as inapplicable to constructions.''

Other quotations from the preface continue to denigrate the mathematics professsion in various ways, so Haupt perhaps would have agreed with Ramsey, while his sentence I called out above belies the understanding he had of the mathematics of his day or any other, as he had a PRACTICAL NEED for consistency in his mathematics.

Later in the same text, Haupt uses the concept of consistency in another way, as follows (see page 24, line 14 from the bottom):

``The condition of equal stiffness, however, does not require that the deflection should be equal for every length: for example, a beam of 20 feet may be allowed to bend twice as much as one of 10 feet, and the expression modified to suit this case will be $\frac{wl^2}{bd^3}$, a constant quantity for beams of equal stiffness.''

In the first part of the above sentence, we find a claim of a relative consistency result, and in the second part, we find a claim which can only be understood mathematically as a theorem about beam stiffness.

But, since I have been alive I have noticed that I can quote all day from books on the theory of bridge making and never convince a doubter that mathematics is relevant to the other disciplines even moreso than the members of other disciplines understand.

Oh yes, and since Ramsey and Haupt are both deceased, I doubt that even an infinite number of days of discussion will convince either of them of anything...

To Juan-Esteban Palomar Tarancon:

Which ``fuzzy logic'' do you claim has ``no room for consistence or non-consistence''. There are infinitely many fuzzy logics. By invoking an ``example'' which has infinitely many disparate members, you give us an empty straw bag against which to pound. By the way, formally, classical logic is a special case of ``fuzzy logic'', in which the lattice of truth values is a (not necessarily two-element) boolean algebra. thus your contention fails for that particular ``fuzzy'' logic.

To Cesar:

A proposed possible proof by G\"{o}del of the consistency of arithmetic is not a proof...

To Cesar:

I did not realize that you have an attachment to your original post. I just found it. It is an article from Paul Cohen, published in 1963: ``A MINIMAL MODEL FOR SET THEORY''. In it, Paul Cohen proves that within every model of set theory there exists a minimal model of set theory. (I'm being a bit vague about WHICH set theory he considered. It was ZF, but that's essentailly irrelevant to the current discussion.) This still is not a consistency proof. It is a relative consistency proof. For in order to construct the kind of model to which Cohen refers, you must have IN HAND a model of set theory.

This article title is misleading, and if Cohen could have made the title less misleading, he probably would have, but the correct title would not have fit on one line.

This situation reminds me of an article by Bob Solovay:

MR0265151 (42 #64)

Solovay, Robert M.

A model of set-theory in which every set of reals is Lebesgue measurable.

Ann. of Math. (2) 92 1970 1–56.

In it, Solovay ``constructs'' a model V of ZF (certainly NOT a model of ZFC!!!) in which the theorem from analysis that there is not a translation-invariant measure on the full power set of the real numbers fails. (In any model of ZFC, the theorem from analysis mentioned above must hold.) His construction begins with a model of ZFC+inacc, that is, he begins by ASSUMING that the axioms of ZFC are consistent, but, aye even more: that there is a model of ZFC in which one can find an inaccessible cardinal number. HOWEVER, the QUESTION of whether the axiom Inacc is consistent with the axioms of ZFC is itself undecidable. Thus, the question of whether Solovay's universe for a set theory in which every set of reals is measurable EXISTS is itself undecidable!! The title of the paper could not have been less misleading, IMHO. But I fault only the width of a piece of paper, not the illustrious Professor Solovay, or the referees or the editors of the Annals of Mathematics.

Peter,

thank you that you came back to my problem with the logical status of school arithmetics. We both see the main point in defining arithmetics by simpler means than those needed in the definition of PA (and not involving torture!)

I once wrote a story, unfortunately in German, which develops such a 'elementary arithmetics system' . The story is at

http://www.ulrichmutze.de/pedagogic_stuff/mms1a.html.

I'll reproduce the relevant definitions here:

Each finite, non-empty string of characters '/' is a natural number (an 'instance of N) and each natural number is such a string. Of course, /// is what we normally write as 3.

Addition of natural numbers is defined as concatenation of strings.

Multiplication is defined by the algorithm which corresponds to the Ruby code

def m(x,y)

xc=x.clone; r=''; r

Peter,

thank you for your thoughts. The problem that systematic manipulations with expressions very easily reaches sizes (i.e. number of characters involved) in excess of the number of Planck cells in the univers is for my 'stone-age coded' numbers not fundamentally different from the situation in mathematics in general. So I take your remark on this as a sign of humor.

Further, you write

There is an answer to the problem of the consitency of Peano aritmethic. The case of set theory is done in an analogous way, God knows when. The answer is given by Reinhard Kahle in a communication with title "Godel's theorem", to be attached to this message.

@Peter:

I don't understand your concern about terminolog, since it is not provable in ZF that being a measurable cardinal is equivalent to being a (strongly) inaccessible cardinal. But, terminology's not a major concern here...

The issue is not the question of theprovability of the existence of such a large cardinal. The reason Solovay's model's existence is debatable is that the relative consistency of the existence of suc a large cardinal is actually not provable.

Where does Goedel prove that we cannot prove the existence of a minimal model of ZF, if ZF is consistent? Please give a reference. I don't think he made such a claim.

I don't know enough type theory to disuss your comments about that. I do hope to someday find time to learn some type theory...

Dear logicians,

would this not be a nice system (probably immune to 'KG's toxic cocktail', Peter)?

The universe of discours consists of just the hereditarily finite sets and our logical system is very much like conventional predicate logic but with quantors restricted from e.g. 'all x' to 'all x which are element of some hereditarily finite set'. The specialization of this idea to arithmetics was already proposed by Peter. The present proposal looks simpler and more homogeneous.

We believe that ZFC is consistent, but there is no proof of that. ZFC is logically incomplete, since the statement "consistency of ZFC", could potentially be proved (if we suppose that does not lead to contradiction) but it not actually provable in this theory. One way to proceed is to attached "consistency" to ZFC and extend ZFC, in a similar way that we attached the solutions of x^2=-1 to the reals and get Complex numbers. Then the question is reduced to the question of consistency of the extended system.

Transitive sets are used to define models of set theory. These models are called transitive models.

In set theory, a set A is transitive iff whenever x\in A and y\in x then y\in A.

The next step is a proof of consistente set theory.

Continuing, set theory is semantically consistent since it has a model.

The work of Kurt Goedel in the 1930s has the consequence that the consistency

of ZF cannot be proved within ZF. So we cannot be sure that the axioms of

ZF avoid some ultimate contradiction.

There are at least two notions of consistency: Syntactic and semantic. We also know that a theory is semantically consistent (known also as satisfiable) if the theory has a model. Indeed ZFC has a model and so it is semantically consistent. Due to 2nd Incompleteness Theorem of Goedel, we cannot transfer this to syntactic side. So the problem remains.

In fact, G¨odel’s Second Incompleteness Theorem implies that it

is unprovable in ZF that there exists a model of ZF. (T. Jech, Set Theory, p. 157)

Proof of Consistency of Set Theory

Quoting [Thomas Jech, Set Theory]

"If M is a transitive class then the model (M,\epsilon) is called a transitive model"

So set theory (i.e. ZFC) is semantically consistent.

Quoting [Volker Halbach, Introduction to Logic]

Theorem(Adequacy)

[Logic L2 is complete and sound]

Theorem(Using the Adequacy Theorem)

A set of L2 sentences \Gamma is semantically consistent iff \Gamma is

syntactically consistent.

By Godel completeness theorem first order logic is complete. Obviously, first order logic is sound.

L2 is first order logic and set theory is semantically consistent iff it is syntactically consistent.

The informal notion of a class needs to be formalized by adding proper axioms to set theory (ZFC), thus this proof is done in an extension of ZFC. This way we avoid the interference of second Godel Theorem.

So we get for free that set theory is syntatically consistent.

Dear Cesar, I think you forget that any formal structure e.g. Set theory that contains natural numbers is incomplete. That;s the 1rst incompleteness Theorem. The second says that: Goedel's second theorem says that, assuming that a certain formal system (ZFC, say) has a certain property that we call "consistency," then there is no formal proof in ZFC of a certain string, commonly denoted by "Con(ZFC).". I thing you confuse the completeness theorem for predicate logic, with incompleteness of First order logic of a formal structure that contains natural numbers. All these are in all books.

Any way if I am mistaking and really you have that proof then you will be famous as tomorrow! Just check again your reasoning.

A theory like set theory can't be simultaneously consistent and complete.

I proved that set theory is consistent so it is imcomplete.

It is known that the comtinuum hypothesis is undecidable in set theory, a result due to Godel and Cohen, showing that set theory is imcomplete, as is known since set theory can formalize arithmetics and by first Godel incompleteness theorem there are formulas which are true but can not be proved.

Dear Cesar,

Suppose you proved the consistency of set theory. And suppose that every "MATHEMATICAL PROOF" can be converted to a formal consistency prof of set theory.

Since almost all mathematics can be coded into set theory, then your proof essentially is the Proof that Hilbert was was trying to prove, and the Goedel's theorems blow up!

What I do not understand is why Hilbert's program prove futile by Goedel's theorems and your "Proof" is OK!

Since you are well versed in these matters, I wish you to be right and me to be wrong!

Dear Costas Drossos,

I turned explicit that the proof of consistency of set theory (ZFC) is done in an extension of it, which formalizes, by proper axioms, the informal notion of a class.

Dear Cesar,

I will be waiting a copy of your published paper!

There are several set theories. This topic was posted some time ago, and I remember that a Czech contributor wrote a link to a document in which a non standard set theory was stated. Likewise, fuzzy set theory is consistent, because under fuzzy logic every contradiction can be avoided by assigning a truth-value less than 1 and greater than 0 to each statement leading to its negation. Recall that fuzzy set membership, usually, is partly true, that is, both relations "x∈A" and its negation "x∉A" can have a truth-value different from false. 

In addition, some contradiction, as the one lying in Russell's paradox can be consequence of the language structure. In fact, those proofs based upon Gödel numbering are language dependent. For instance, consider the formal expressions "2", "|√4|", "(1+1)" , "2+0". These expressions possess the same meaning "2"; however any Gödel numbering function assigns different integer to each of them. Accordingly, Gödel numbering works with notations, formalisms or languages instead of concepts or mathematical constructions. In other words, any proof based upon any Gödel numbering not only is derived from the assumed axioms and logical rules, but the language structure too. In general, metamathematics is language dependent. Notice, that the contradictory concept of "set of all sets" is avoided by means of a language machinery: Such a collection is termed class instead.



Finally, I would like to remember the following quotation.

"Suppose a contradiction were to be found in the axioms of set theory.
Do you seriously believe that a bridge would fall down?"
Frank Ramsey (1903 - 1930)

@Cesar

Where can I find your proof ???????????????

Dear Cesar, just a final note. The Goedel's Theorems, say that it is impossible to prove the consistency of ZFC, with means of ZFC. Now if you make an extention of ZFC, then using the tools of this extension you might prove the consistence of ZFC. However there is a problem: This proof is depended on this extention. How do you know that this extention is secure? Otherwise your proof will suffer to. Along ago I was thinking that I may use a nonstandard or Boolean extention of ZFC, to takle the consistency of ZFC. You should first prove that the extention is consistent if ZFC is. This is a program which tries to escape Goedel's theorems and at the same time you can prove a relative consistency of ZFC. Good luck!

Dear Costas Drossos, the second Goedel theorem says that for a special deductive sytem T, from T we can not deduce CON(T), the consistency of T.

In the proof from ZFC+, the extension of ZFC that treats with classes, we deduce the consistency of ZFC. The second Godel Theorem is respected. In the meanwhile I made a publication with title "A Proof of Consistency of Set Theory (ZFC)", corresponding to the proof presented above.

Cesar, again, where can I find 'the proof presented above'?

@Cesar Rodrigues. OK Cesar, I agree that proving CON(ZFC) from some extension ZFC+, it is OK, and Goedel' s theorem is respected. However, the result depends on ZFC+. Is this consistent? Otherwise the proof looses its importance. Any way, publish this result and I am sure that during the refereeing process it will get better.

@ulrich The proof is present in a publication with title "A Proof of Consistency of Set Theory (ZFC)".

Cesar, a publication is something that has been published. So you should be able to give a pointer to it. Does 'presented above' point to heaven ? If you have written the stuff down you could easily put it to your RG publications and we could avoid inconclusive discussions as they presently are taking place.

@Cesar. I downloaded you paper "A Proof of Consistency of Set Theory (ZFC)", and I did not like at all. Thus what your proof means, I think you just repeat the relative consistency result of Goedel, and this is not new. The general consistency problem remains without proof.

Let me give you some rational for that.

Since ZFC is incomplete (There are independent statements), one can not use the usual completeness theorem of first order logic to infer completeness from the inner model of ZFC.

In fact Goedel constructed an inner model using constructible sets, and in addition L satisfies AC, and CH. Thus there are no independent statements and he can use the completeness Theorem to get the first relative consistency in set theory via “inner models”.

Thus what your proof means, I think you just repeat the relative consistency result of Goedel, and this is not new. The general consistency problem remains without proof.

In Jech, p. 157, we find:

Goedel’s Second Incompleteness Theorem implies that it

is unprovable in ZF that there exists a model of ZF (Jech ,157).

As for your extension using classes, I do not think that this a formal extension like NGB. Classes, e.g.. see Jech, p. 5, are defined and also operations etc. but this does not mean that we have classes in ZFC. Classes are only a notation|-we can always eliminate their use.

See also, http://mathoverflow.net/questions/22635/can-we-prove-set-theory-is-consistent

I forgot to note that the inner model of Goedel includes also the axiom of constructibility: V=L. This radically restricts the "sets" and it is a problem if this restriction is compatible with the conception we have for sets. Doing mathematics in ZFC+VL, it is not sure that covers all the needs for mathematics. Thus your proof does not prove that mathematics are consistent. Usually we use von-Neumann universe for doing mathematics. But due to incompleteness we can not prove that there exist a model!

@Costas, the extension to ZFC, you called ZFC+, could be set theory NBG, a conservative extension to ZFC.

We don't know if NBG is consistent, but that ZF and NBG are equiconsistent.

Nowadays, we don't ask the absolute consistency of a theory, but we try to define its relative consistency to other theory, or equiconsistency as above.

@Cesar: OK! This is a minor issue. The big issue is that the inner model of set theory based on constructible sets, has also as an additional axion V=L. This reduces the concept of "set" that intuitively have. Any way, this is not a model of ZFC without any additional axiom. In fact since ZFC have independent statements, does not exist any model. Thus semantically completeness of ZFC does not exist.

My idea I was told you in a previous post is to consider a nonstandard version of ZFC and see if you can prove Con(ZFC) considered as an internal object. This may also not work.

In particular, there are variants of NF (New Foundations) [W. Quine, 1937] set theory which are consistent, e.g. NFU (NF with urelements) [R. Jensen, 1969] which is:

. Consistent;

. Consistent with the axiom of choice;

. Consistent with the axiom of infinity;

. More intuitive than ZF.

NFU can "safely be extended as far as you think ZFC can be extended".

Dear Cesar,

OK, lets accept that NFU have all these properties. There are tow ways to go from here: Either you accepts NFU, and forget about ZFC. of prove that the consistency of NFU implies the consistency of ZFC. Then you will have a relative consistency proof.

Of course you have to check if some body else have such a proof or your proof is original.

By now we can continue using ZFC because no inconsistency emerged. If any problem appears in the future with ZFC, we can replace it with its alternative, i.e. NFU. The following communication presents the evolution of NFU during 40 years since 1969.

I realized that a slight change of the proof of relative consistency of ZFC relative to NBG, gives the proof of relative consistency  of ZFC relative to NFU+infinity+choice.

In the original proof NBG was employed to define classes. NFU+infinity+choice is consistent, and can also be employed to define classes, including proper classes.

If NFU+infinity+choice is consistent, than by the respective relative consistency  result, it can be proved that ZFC set theory is consistent.

An important reference to NFU is [R. Holmes, 98].

A dataset containing this proof and the proof of the relative consistency of ZFC relative to NBG set theory, has been added to my profile.

Dear Cesar,

Now your talking OK!

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

https://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf

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.

Very well said Mihai!

Thank you Costas, I also appreciate the discussions initiated by you and your ecouragements.

NF (New Foundations) is a set theory with many consistent variants, including set theory NFU, NF with (Ur)elements. NFU is consistent as the result of its relative consistency relative to Peano arithmetic, which is consistent (see the  proofs by Gentzen and by Goedel).

NFU+infinity+choice is a consistent variant of NFU, as can be seen in the NFU literature.  In the same way ZFC is consistent as a result of its relative consistency relative to NFU, which is  consistent. This talk began in the last century in the year of 1969, with the work of Jensen who proved that NFU is consistent.

The proof in answer 70 is replaced by the following proof, it is intended to mean.

In the context of NFU+infinity+choice set theory, which is consistent,   we define a transitive class M, which defines a transitive model  of set theory.

So set theory is semantically consistent and syntactically consistent too, as it is a first order logic theory.

All in all, we proved that if NFU+infinity+choice is consistent then set theory is consistent, and that set theory is consistent since NFU+infinity+choice is consistent. 

Akira,

please educate us with respect to the consistency of Peano arithmetics. Is'nt it a Goedel theorem that all theories which contain Peano arithmetic can prove their consistency only if they are inconsistent? What then is the basis of Gentzen's and Goedel's proof of consistency of Peano arithmetics?

Dear Akira,

Your first question seems very interesting! I had some years ago similar ideas. Can we considered a nonstandard model of arithmetic as an externa; model to standard mode; of arithmetic? If we cannot prove consistency inside the standard model of arithmetic, using nonstandard model we have an external model which whoever restricting to internal entities looks the same as the "standard one"!

Costas:  "...looks the same..." -- And we are back to the hair-pulling circularity of Cantor set's cardinality, local and global self-similarity, if ever to be well-founded as Godel suggests is to be well-founded outside mathematics entirely, subtly in the physics of nature.

The proof of consistency of set theory in (R)esearch (G)ate technical report, i.e. dataset,

"Consistency of Set Theory from the consistecy of NFU set theory"

is a simplification of answers 70 and 76 (the proofs of consistency of set theory in this question). 

Akira, any text on classical mechanics would show you how GMm/r^2 when properly applied to a n-particle system (e.g. n=2) gives rise to a Galilei-invariant theory in which for each pair (i,j) of distict particles action equal reaction and there is a well-defined relative velocity vij such that vij=-vji. What you write on GMm/r^2 is ridiculous. For me it is sad to see that mastering some mathematics does not prevent people from completely misunderstanding physics.

Dear Akira,

I do not know if you have seen the book: [Anna_Horská]_Where_is_the_Gödel-point_hiding. It is available if you needed.

Akira,

in my note I intended to point out that exactly what you want me to write down can be read in many textbooks on classical mechanics so that I don't need to annoy anybody with self-evident material . Since you took my words for something different here is my direct explanation. For ease of writing we consider only 2 bodies and assume that the velocities of both are along the line which connects the bodies. Actually we assume this to hold for one point in time and as a consequence of this it will hold for all times.

Let us write x and X for the positions on the line of motion (in Newton's absolute space). Then the rule for which 'GMm/r2' is a short form is to let mass m feel the force (X-x) mMG/(x-X)³ and the mass M feel the force (x-X) mMG/(x-X)³. Writing a and A for the accelerations, we thus get a=(X-x) MG/(x-X)³ and A = (x-X) mG/(x-X)³  and thus ma+MA=0 which is actio = reactio. Of course the accelerations are relative to Newton's absolute space. Writing time derivatives as ' (instead of dot) we have the differential equations x''=a and X''=A  which define (together with initial values for x,X,x',X') trajectories x(t) and X(t) and thus a relative position r(t) =x(t)-X(t) and a relative velocity x'(t)-X'(t). Changing the roles of x and Y changes signs but conserves the absolute values. Where is the contradiction? In the reputed textbook by Herbert Goldstein this is discribed in detail in Section 2 of Chapter 1 (at least in my German edition from 1963). So this not my definition of Newtonian mechanics (which you asked for) but the textbook definition.

I can't but blaming you for using strong words against scientific communities in a situation where a short conservation with any normally educated theoretical physicist had freed you from your severe misconception.

Oh Akira, do you want to start one more crackpot discussion on Special Relativity in Research Gate? I expected that you will give your reasons to derive from GmM/r² equations of motion which differ from the textbook version. (Aha, you probably wrote your last note not having read mine, so perhaps do not exspect in vain).

Akira,

My note beginning "in my note I intended to point out ..." is my answer to your

It is not related with set theory.

@Mule

You are completely right, this is not related to set theory. It grew however out of discussion of the central topic in an inevitable way since Akira Kanda, after having given valuable contributions to this discussion, went on with the following paragraph:

m. I do not think that those who can not see this contradiction will really understand what Godel did. Considering the anti-logic stance of the "theoretical physics", I do not think there is any who understand that this problem.>>

I thought that leaving such strong oppinions unchallenged would leave some readers, who so far worked with Newton's law of gravitation without becoming aware of logical flaws, in an uncomfortable mood.

Note that a consevative extension of a consistent  theory, is consistent. For instance,  set theory NBG is a conservative extension of set theory, a consistent theory. Thus NBG is also consistent.