Talk:Zermelo–Fraenkel set theory/Archive 1

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Some early questions

I'm confused: are the listed axioms for ZFC or von Neumann thingy lala? -Martin

They're ZFC; I've clarified. As a separate point, isn't the empty set axiom redundant here? It seems to follow from infinity and replacement. Matthew Woodcraft

It is redundant in some formulations, but not others. In any case, it's traditional to include it. I'll mention the redundancy on Axiom of the empty set. -- Toby 05:33 Feb 21, 2003 (UTC)


The singular, "Zermelo-Fraenkel axiom", does not make sense as the title of this article. It makes more sense to title an article "cat" than "cats", and is in accord with Wikipedia conventions, but we're not defining a general concept of a Zermelo-Fraenkel axiom; we're defining a short list of specific axioms and schemas; the whole phrase "Zermelo-Fraenkel axioms" is really a proper noun. It's as if ten separate articles were titled "Comandment" without any article titled "Ten Commandments". The plural in the title of this article makes sense for the same reason the plural in an article titled "Ten Commandments" would make sense. Michael Hardy 22:53 Jan 15, 2003 (UTC)

How about Zermelo-Fraenkel set theory? I'm finding that in metamathematics books more than anything else. -- Toby 05:33 Feb 21, 2003 (UTC)


Do most mathematicians believe anything about ZF? Most mathematicians use operations on sets, and the validity of those operations in in effect codified by ZF, but I don't think most mathematicians think about ZF, let alone believe anything about ZF. Michael Hardy 23:08 Jan 15, 2003 (UTC)

I agree; most mathematicians couldn't care less about ZF. Sure, the axiom of choice is interesting, but not the axiom schema of replacement or the axiom of well foundation. I've changed it to "metamathematicians", which may not be precisely the right term. -- Toby 05:33 Feb 21, 2003 (UTC)


"On the other hand, the consistency of ZFC can be proved by assuming the existence of an inaccessible cardinal." Does this refer to weakly or strongly inaccessible cardinals? -- Schnee 01:25, 10 Aug 2003 (UTC)

Weakly -- all inaccessible cardinals are weakly inaccessible, while only some of them are strongly so. -- Toby Bartels 22:33, 13 Feb 2004 (UTC)

Just noticed this comment; it's a couple years old, but should probably be addressed for the record. Actually the more common convention is that "inaccessible" means "strongly inaccessible", and if you mean "weakly" you say it explicitly, unless it's clear from context. However from the point of view of Schneelocke's question, it doesn't matter, because the existence of a weakly inaccessible cardinal has the same consistency strength as the existence of a strong inaccessible. --Trovatore 02:40, 29 April 2006 (UTC)

Shouldn't we state the axioms in their weakest forms, i.e. "if two sets are the same then they are equal" and "there is a set"?

list of set theory topics

Wikipedia has no list of set theory topics! Set-theory mavens, please help. Once it is created (or maybe even before it is created?), it should be added to the list of lists of mathematical topics. Michael Hardy 00:30, 13 Jun 2005 (UTC)

What does the exclamation point mean in logic?

The first order logic articles doesn't list it as one of the symbols, yet it is this article. --212.85.24.83

means "there exists a unique". --Zundark 15:47, 6 December 2005 (UTC)

Consistency proof (partial, of course)

http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html reads: "Abian (1969) proved consistency and independence of four of the Zermelo-Fraenkel axioms". The original paper is available at http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.ndjfl/1093888220. I would really love to add a note about it to this article, but I fear I lack the necessary understanding on the topic.


Is the axiom of infinity consistent with the axiom of foundation?

Basically, we start with the set of all ordinals that do not contain themselves. This set, just like the set in Russell's paradox, is not well defined.The question that comes up is whether it is the set N of all finite ordinals or the set of Burali-Forti paradox.The following reasoning makes me think that it is the set N itself.

For, N={0}U{Sx:x eN},

This is true, but it is not the definition of N.--Aleph4 13:48, 27 January 2006 (UTC)

where S is the successor function ('e' is 'belongs to';! prefixed is the negation).By the axiom of foundation, x !e x for all x.

So, N is {0} U {Sx:x e N and Sx !e Sx}.

But {Sx:x e N and Sx !e Sx} ={y:y!=0,0 e y,y e N,and y !ey}

[The above equality holds only because N is infinite]

So, N={0}U{y:y!=0,0 ey,y e N,and y !ey}.

But the condition y e N is tautological.

No, it is not.--Aleph4 13:48, 27 January 2006 (UTC)
Please see the clarification below.--Apoorv1 06:59, 2 February 2006 (UTC)

So,N={0}U{y:0 e y and y !e y}.

But this set is not well defined.

See also links http://mathforum.org/kb/message.jspa?messageID=3808877&tstart=0 and http://mathforum.org/kb/message.jspa?messageID=4165416&tstart=0). See also[[1]]

--Apoorv1 06:55, 27 January 2006 (UTC)

Those are cryptic comments. A little more elaboration would help.--Apoorv1 05:16, 30 January 2006 (UTC)

In the absence of any further elaboration from your side, these are some additional comments.

'When we set

N={0}U{y:y!=0,0 ey,y e N,and y !ey}, it means

y e N iff y=0 or [y !=0 and 0 e y and y != y and y e N].

Now, for statements p and q, [p<-->p and q]<-->[p<-->q],so

I guess you mean [p <-->(p and q)] in the first bracket. Your formula is not a tautology. The first bracket is equivalent to [ p --> q ]. --Aleph4 10:28, 2 February 2006 (UTC)
The reverse implication is also true in this case, because the axiom of infinity by itself does not guarantee the existence of a set larger then N all of whose member contain 0.. So, in the system 'ZF less the powerset axiom', N is the set of all ordinals containig 0 that do not contain themselves. Alternatively, it is the set of all ordinals containing 0. In any case,in the system 'ZF less the powerset axiom'the set guaranteed to exist by the axiom of infinity will not satisfy the axiom of foundation.If the system 'ZF less the axiom of powerset' is inconsistent, will the system ZF be consistent?--Apoorv1 07:34, 6 February 2006 (UTC)
What do you mean by because the axiom of infinity by itself does not guarantee the existence of a set larger then N all of whose member contain 0.? In particular, what do you mean by "larger"?
  1. ZF minus Power set does not guarantee the existence of an uncountable set.
  2. ZF minus Power set does guarantee the existence of a set X with the following properties:
  • X contains N
  • X is not equal to N
  • all elements of X (except for 0) contain 0.
(Note that in the context of set theory, N is always understood to contain 0. I hope I have not misunderstood you there.)

Which precisely is this set X that you are referring to ?----Apoorv1 04:59, 18 March 2006 (UTC)

For example, the set omega+1.--Aleph4 08:21, 28 March 2006 (UTC)

X is supposed to be closed under the successor operation. So how is it w+1?In fact, as I have averred earlier, the repeated application of the successor operation can give you no set bigger than N, unles you assume that N !e N, or equivalently S(N) !=N.In the absence of the axiom of powerset and regularity, the membership of N in N cannot be decided.--Apoorv1 08:55, 28 March 2006 (UTC)

There is another way of approaching the issue. As you say,

  1. ZF minus Power set does guarantee the existence of a set X with the following properties:
  • X contains N
  • X is not equal to N
  • all elements of X (except for 0) contain 0.

This means, that the axiom of infinity is actually two different axioms:

1A)The set N, containing 0 and closed under successor operation exists.

1B)Another set X, containing 0 and N and closed under successor operation exists.

Once again consider the system ZF minus powerset and regularity and only 1A part of the axiom of infinity.

Then N is the largest ordinal in this system and hence, N eN <--> N!eN .

The addition of axiom 1B or the axiom of regularity or powerset does not help us to resolve this basic contradiction.

--Apoorv1 11:14, 31 March 2006 (UTC)

N is never defined to be the set of all ordinals containing 0 that do not contain themselves. The clause "do not contain themselves" does not make much sense, because whenever x is an ordinal, then x is not an element of x. This follows from the definition of "ordinal".
If the system 'ZF less the axiom of powerset' is inconsistent, then of course ZF is also inconsistent. But you have not shown either of the two statements.
Aleph4 20:45, 16 March 2006 (UTC)

See remarks above.--Apoorv1 06:01, 28 March 2006 (UTC) I moved them to "below" Aleph4 08:21, 28 March 2006 (UTC)

y e N iff y =0 or [y !=0 and 0 e y and y !e y],so

N={0} u {y : 0 e y and y !e y} .

The analogy with

S={y:y e S and y=2} <-->S={y : y=2}

makes the above reasoning clearer'.--Apoorv1 06:59, 2 February 2006 (UTC)

The empty set will satisfy the left equality, but not the right one. --Aleph4 10:28, 2 February 2006 (UTC)
The point is well taken. However, see the comments above.--Apoorv1 07:34, 6 February 2006 (UTC)

Infinite sets

The discussion was getting a bit confusing, so I moved Apporv's question here.Aleph4 08:21, 28 March 2006 (UTC)Just to ensure readability, I have copied the relevant comments of Aleph4 below.--Apoorv1 10:50, 28 March 2006 (UTC)

'ZF minus Power set does not guarantee the existence of an uncountable set. ZF minus Power set does guarantee the existence of a set X with the following properties: X contains N; X is not equal to N;and all elements of X (except for 0) contain 0.' Aleph4

For the moment, let us say that X is some (as yet unidentified) countable set.Now X is infinite only if Sx!=x for all xeX.But Sx!=x iff x!ex.So X ={0 and all x containing 0 that do not contain themselves}. Now consider the system ZF less the powerset and regularity axioms.In the absence of the powerset axiom, P(X) does not exist. S(X) exists only if X !e X. But X is nothing but the set in Russell's paradox and the question whether X e X or X !e X cannot be answered.So X, which is countable by hypothesis,is not well defined. --Apoorv1 06:01, 28 March 2006 (UTC)

It seems that you are claiming various things that you cannot prove. For example, "Now X is infinite only if Sx!=x for all xeX". It seems to me that you are claiming
(A) if X is infinite, then Sx !=x for all x in X.
Or perhaps you meant to say
(B) if Sx != x for all x in X, then X is infinite.
I think we can agree that (B) is false. (e.g., take X empty).
The axiom of regularity implies that Sx != x for all x (because x not in x, for all x). So if you assume regularity, the clause "if X is infinite" is redundant.
Without the axiom of regularity, (A) cannot be shown.
Aleph4 08:21, 28 March 2006 (UTC)

I think we need to see my comments in the context of our discussion.The set X we are talking of is the set guaranteed to exist by the axiom of infinity and is closed under the successor operation. If X is finite, it could not be closed under the successor operation. If X is infinite and Sx=x for some xe X, then x =X and so X e X and SX=X for a countable set, directly in contradiction to the axiom of regularity.

So the only case of interest is the case when Sx!=x (i.e x!ex)for all x in X.

In this case, in the system ZF less powerset and regularity, the set S(X)!=X only if X !e X.Since X = {All x such that x !e x}, X , which is countable, is not well defined.Since X !e X <-->X e X in this system, the addition of the axiom of Regularity to this axiom system cannot remove this basic contradiction. --Apoorv1 10:35, 28 March 2006 (UTC)

Empty Set, Pairing, Subsets are redundant

I draw your attentions to the masterly exposition of Suppes (1972). Suppes dispenses with Empty Set by simply deriving the empty set as the extension of A not equal to A. He then sets out Pairing, and Subsets (Separation) very early on, and delays introducing Replacement as long as possible. But when he does so, he shows that Pairing and Subsets become easy theorems. Thus his definitive listing of the ZFC axioms, on p. nn, does not include Empty Set, Pairing, and Subsets.

It is indeed a revealing fact that the vast majority of working mathematicians don't know any axiomatic set theory and are not curious about it. They are not even interested in the foundations of mathematics. Taking set theory seriously seems limited nowadays to Berkeley, Tarski's students, a number of Israeli and Eastern European mathematicians, and the coterie studying Quinian set theory. The limited interest in set theory and metamathematics nowadays may be largely driven by the lack of interest in those subjects on the part of granting agencies.202.36.179.65 18:00, 27 February 2006 (UTC)

One might say that Paul Cohen killed the foundations of mathematics as classically conceived by answering (in the negative) in 1964 what remained post-1950 as its biggest open problem, whether Choice and GCH followed from the ZF axioms. Thereafter foundations went off in two directions. One direction, the primary outlet for which was then and still is JSL, the Association for Symbolic Logic's Journal of Symbolic Logic, continues to address the progressively more esoteric questions remaining within the ZF framework, of which there are plenty but which the average mathematician finds it harder to relate to as more of them get answered. The other direction is comprised of various new and not so new frameworks whose respective perspectives make the questions raised by classical foundations less well motivated and which instead substitute their own open problems motivated by their own perspective. Proof theory, modern or abstract algebra with an emphasis on universal algebra, and category theory are all active subjects today, each with at most one or two hundred actively contributing participants, each considering itself as addressing foundational concerns in mathematics. But even those are becoming old hat, and today we find a lot of interest in foundational studies of coalgebras (which arose from categorical thinking but which could as well have come from ZF), quantum programming languages (as a much needed sensitization of Birkhoff and von Neumann's old quantum logic to the more quantitative and dynamic aspects of Heisenberg uncertainty and entanglement), concurrency theory (broadly construed to cover any kind of concurrent behavior by fleets of vehicles, packet networks, parallel programs, corporations, orchestras, armies, etc.), and so on. Foundations is far from dead, it just isn't recognizable today if you define it narrowly to be ongoing research into the consequences of the ZF axioms. In view of this diversity, to say that the ZF axioms are the axioms on which mathematics is based today is a bit of a stretch. Sets and functions play a very important supporting role in modern mathematics, but the ZF axiomatization of binary membership, however popular with old-school foundationalists, is by no means the only approach to either foundations in general or sets and functions in particular. The language of ZF is not even mathematically natural: when did you last see anyone make use of a homomorphism that respected set membership? Monotone functions respect order, group homomorphisms respect the group operation, linear transformations respect linear combinations, and gangsters respect membership in the Cosa Nostra, but what morphism has ever respected membership in a set? It is sheer hubris for a relation that can't get no respect to claim to support mathematics. Vaughan Pratt 00:59, 21 August 2006 (UTC)

Right arrow

What does the mean ? Where is it defined ? Thanks. --Hdante 08:44, 5 March 2006 (UTC)

The right-arrow is the symbol for material implication in propositional logic. See Propositional logic#Soundness and completeness of the rules. Otto ter Haar 12:24, 5 March 2006 (UTC)

Symbolism

Where is the cheat sheet to explain what all the symbols mean? Kd4ttc 22:12, 10 March 2006 (UTC)


For those of use familar with mathematics but not expert the symbols in the text are opaque. Any reference that one can go to for the symbol meanings? Kd4ttc 23:02, 12 March 2006 (UTC)

Does first-order logic help? --Trovatore 23:14, 12 March 2006 (UTC)

Yes! I'm thinking of developing a compendium of symbols that the casual reader may browse. Kd4ttc 23:40, 12 March 2006 (UTC)
So there's already Table of mathematical symbols and List of operators (these are probably duplicative as it is). You might take a look at the best way to help people find these. --Trovatore 23:44, 12 March 2006 (UTC)
You are more than kind! Kd4ttc 01:12, 13 March 2006 (UTC)

Syntax and semantics

At the moment the article has syntax and semantics all mixed up. ZFC per se is purely syntax; a collection of strings of characters and rules for manipulating them. Therefore, for example, the first sentence from the introduction,

ZFC consists of a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (i.e., all mathematical objects) are sets.

is wrong; ZFC, in and of itself, has no ontology. The set-theoretic notions that (depending on your philosophy) interpret, justify, or motivate ZFC, do have an ontology, but they are distinct from ZFC itself.

(Really it's these set-theoretic notions that should be given the status of "the most common foundation of mathematics", not ZFC itself.) --Trovatore 03:41, 12 March 2006 (UTC)

Are ordinals essential?

I changed the sentence (referring to Zermelo's axioms)

This axiomatic theory did not allow the construction of the ordinal numbers, and hence was inadequate for all of ordinary mathematics.

because it is not true. Most of "ordinary" mathematics can be developed without using ordinals. Transfinite induction does appear sometimes in "ordinary mathematics" (which I understand in this context as "mathematics except set theory", or perhaps "mathematics except mathematical logic"), but usally any well-ordered set of the appropriate length will do, and in fact transfinite induction is usually done along any well-order of the set that is being investigated at the moment (field whose closure is to be computed, Banach space on which a functional is to be extended, etc). Often even transfinite induction is not used, and is replaced by using Zorn's lemma as a black box.

Set theory of course all the time uses von Neumann's cumulative hierarchy, where ordinals and the replacement axiom are quite natural or even necessary.

Aleph4 20:54, 16 March 2006 (UTC)

The silly consistency comment

I refer to the following comment

Because the axioms of Peano arithmetic are ZFC theorems, and the consistency of Peano arithmetic cannot be proved by virtue of Gödel's second incompleteness theorem, the consistency of ZFC cannot be proved using ordinary mathematics.

On its face, this sentence is misleading at best, false at worst. There are many known consistency proofs of PA, Godel's theorem notwithstanding. I plan to fix this soon. CMummert 22:37, 26 April 2006 (UTC)

Yet Another Silly Consistency Comment?

"Because of Gödel's second incompleteness theorem, the consistency of ZFC cannot be proved within ZFC itself. " Surely this should read something more like "Because of Gödel's second incompleteness theorem, the consistency if ZFC could be proved within ZFC itself, then ZFC would be inconsistent"?

Of course, ZFC is consistent, but we don't *know* that (do we?). SinghAgain 17:14, 04 Feb 2007 (UTC)

Distinctness condition missing in formal statement of Axiom of Choice?

Shouldn't there be an assertion that D != B in the statement of the axiom of choice? That is, two _distinct_ elements of A are disjoint. Otherwise, B and D can be the same element of A, in which case the consequent of the innermost implication fails and the axiom as a whole is vacuous. Or have I missed something...? Awmorp 11:33, 28 April 2006 (UTC)

Yes, you're right. CMummert 11:37, 28 April 2006 (UTC)

Limit Ordinals-a matter of faith?

The first limit ordinal is w (omega).It is defined as the union of all preceding ordinals. This presupposes that the set of all preceding ordinals exists-so the definition of w really presupposes the existence of and the definition of w (and is a non definition).

If we consider numbers as primary, then we know that given a number, there is a greater number,but there is no infinite number.Similarly,given a finite set of numbers, we can have a bigger set.But can we have an infinite set?

If we consider the seq. A1={1},A2={1,2},A3={1,2,3} and so on,what is the process by which the limit {1,2,3. . .}is obtained? In particular, there is no metric by which the successive members become closer to each other.So the axiomatic assertion,through the 'axiom of infinity' of the existence of a 'biggest' or 'infinite set w' appears to be a leap of faith,which is quite opposed to our assertion that an infinite number or magnitude does not exist. --Apoorv1 07:47, 12 May 2006 (UTC)

Well, an axiom basically is a formal leap of faith. You can prove that the axiom of infinity cannot be proven or disproven from the rest of ZFC. -Dan 17:11, 24 May 2006 (UTC)

Nitpick about finite axiomatizations

...proved that ZF (and hence a fortiori ZFC) cannot be ... finitely axiomatized.

Why a fortiori? Maybe this is backwards? -Dan 17:11, 24 May 2006 (UTC)

Good point. Anyone want to look up Montague's paper, and see what he actually proved? (We shouldn't reverse them unless he really did prove ZFC not finitely axiomatizable in 1957). --Trovatore 22:27, 24 May 2006 (UTC)
ZFC really is not finitely axiomatizable, because it proves that any finite subset of its axioms has a model. The sentence in the article is misleading. CMummert 11:40, 25 May 2006 (UTC)
I have now fixed that sentence and the false statement about Godel's theorem that I mentioned higher on the talk page. CMummert 12:00, 25 May 2006 (UTC)
Of course I knew that ZFC is not finitely axiomatizable. My question was about what Montague had proved. While it seems most natural that he would have proved the result for ZFC, I haven't actually seen his paper. Have you? --Trovatore 13:53, 25 May 2006 (UTC)
No, I haven't; I left in the reference to Montague only out of respect for the original author, and I would not mind if it were removed. I did look up some papers on Mathscinet before I edited the article this morning. The best bet seems to be MR0163840, which is dated 1961 instead of 1957. Here is a quote from its description of Montague's paper Fraenkel's addition to the axioms of Zermelo with ellipses to indicate where I pruned it.
Fränkel's addition is the replacement (or Ersetzungs-schema (RS). ... The author defines a (countable) subset of the cumulative type structure which can be proved to satisfy all instances of Zermelo's comprehension schema (SSF: schema of set formation); ... And if an additional finite set $A$ of sentences in the notation of set theory is added, a model satisfying both $A$ and SSF can be established. ... A consequence is that RS is not finitely axiomatisable over SSF, and the same holds for any consistent extension of RS ...
That quote indicates to me that Montague did prove the result that the article indicates, although the year may be wrong. Moving to the area of personal opinion, I generally feel that there is little reason to give attribution of results such as this one in wikipedia. Extremely important or extremely difficult results may deserve special attention, but this result does not have those properties. So I would vote in favor of not attributing the result at all in the article, and just pointing out the ZFC is not finitely axiomatizable. CMummert 14:36, 25 May 2006 (UTC)

Switch to Kunen's axioms

I changed the previous set of formal axioms, which I think were correct, to the exact set of axioms in Kunen's book. Here are my reasons:

  • Kunen was the first book I could find that gave symbolic forms of the axioms.
  • The other set of axioms was unsourced.
  • The other set of axiom was typeset poorly. The conventions were not those employed in contemporary literature. Several of the previous axioms were typeset at over 8 inches of width on my monitor. I could not easily tell whether the previous axioms were correct because the typesetting was too hard to read.

I also added references to Kunen and Jech's books. Once I figure out how to do proper inline citations I will fix that. I think that this article should somewhere mention the cumulative hierarchy, which is the fundamental motivation for the axioms of ZFC. CMummert 20:38, 23 June 2006 (UTC)


I have several objections concerning some details in Kunen's list:

  1. The axiom of set existence is not really set-theoretical; it is a purely logical axiom (or a consequence of purely logical axioms). I suggest to either omit it altogether (as some axiomatizations do -- e.g. Jech or Fraenkel, Bar-Hillel, Levy), or to replace it by the "axiom of the empty set". The axiom of the empty set follows of course from the axiom of separation, but one needs the axiom of the empty set to deduce the axiom of separation from the replacement axiom.
  2. I think that Kunen is unique in calling the "well-ordering theorem" an axiom. He does this only for his own convenience to speed up the development in his book. There are many theorems that are (over ZF) equivalent to the axiom of choice, but only few of them deserve the name axiom (rather than "theorem" or "lemma"). I suggest to use one (or several) of the customary formulations of AC (choice function for P(X), choice function for families of nonempty sets, choice function for disjoint families/disjoint sets of nonempty sets).

--Aleph4 18:39, 24 June 2006 (UTC)

I have made some changes that at least partially address your concerns.
A disadvantage of using Kunen's axioms would be that we are stuck with using them more or less exactly as he phrased them; if we change them, then they aren't Kunen's axioms any more. I did look at Jech's book; he gives the axioms in English, but not in symbolic form. I think it is nice to have a concrete reference for the specific formal axioms that we list, since the axioms are not completely canonical. It seems to fit WP:NOR better. On the other hand, I added English decriptions based on the previous article, and pointed out that those descriptions are not from Kunen.
I have no objection if someone else finds a book that gives formal symbolic statements of the axioms, puts those symbolic statements into the article correctly, and gives a correct citation to them. That is, I am not advocating Kunen over any other source. Kunen's book was just the first book I could find, and I thought it was sufficient.
I think that it would benefit the article more to explain the various ways in which the axioms are not canonical than it would to pick a different noncanonical choice of axioms.
My impression of the distinction between the well ordering principle and the axiom of chice is this. Cantor suggested the well ordering principle (Every set can be well ordered) as an axiom in the 1880s. Some opposition to the supposed obvious nature of this axiom arose, and Zermelo gave a proof which reduced the well ordering principle to the axiom of choice (Every sequence of nonempty sets has a choice function), which Zermelo believed to be conceptually simpler. So the well ordering principle has historical precedent as an axiom.

CMummert 19:14, 24 June 2006 (UTC)

As I understand history (although I do not have a reference at the moment), Cantor did not consider an "axiomatization" of set theory at all. (Does your "1880s" refer to "Über unendliche lineare Punktmannigfaltigkeiten"?)

On the other hand, the main point for Zermelo's 1904 and 1908 papers was to prove the well-ordering theorem as a theorem, and to isolate the axioms used in this proof, in particular the axiom of choice.

Jech's book (millenium edition) gives the axioms in English on the first page, and a few pages later gives formal versions. I like these version better than Kunen's -- not only because the axiom of choice is given in the customary form, but also because he uses lowercase and uppercase variables, which is more intuitive.

Aleph4 19:50, 24 June 2006 (UTC)

Ah. I saw the English ones at the start of Jech's book, looked at Kunen's book, and stopped looking. I have no objection if you would like to switch the article to Jech's axiomatization. CMummert 20:32, 24 June 2006 (UTC)


There are two different types of bi-implication arrows used in the Axioms as set out here. Is there a reason for this or should it be changed? 86.20.228.25 16:12, 4 February 2007 (UTC)
I cleaned that up some; it had drifted since the version originally added. I have no preference at all for Rightarrow vs. rightarrow, so I just chose one. You should feel free to make corrections like this yourself. CMummert · talk 18:03, 4 February 2007 (UTC)

Axiom of extensionality

The English description for this axiom reads "Two sets are the same if and only if they have the same members." But in fact, the axiom as it is presented states that "Two sets are the same if they have the same members" (a conditional rather than a biconditional). I'm not familiar with the original text from which this axiom was taken, but it's presented as an iff in the stand-alone article (axiom of extensionality). Whichever we decide to use on this page, the English description should match the mathematical description. Mathfreq 21:59, 15 August 2006 (UTC)

The fact that sets that are equal must have the same members is a property of equality which is an axiom of the underlying first order logic. Thus only the converse, that sets with the same members are equal, needs to be added as an axiom. More importantly, the formal axioms are directly quoted from Kunen; please don't change them unless you have a reference for the new ones. Anyone is free to change the English text, which is original here, to explain what is going on. CMummert 22:44, 15 August 2006 (UTC)

Axiom of Choice v. Well-ordering Theorem (v. Zorn's lemma)

I'm just wondering why we've labeled the well-ordering theorem with its ZF-logically equivalent "axiom of choice". The well-ordering theorem applies terms such as "minimal" which have no defined context.

Secondly, these axioms are supposed to allow an intuitive basis for ZFC logic. We very well could substitute the axiom of choice/well-ordering theorem with Zorn's lemma. This creates a structurally identical system; however, it would create more confusion.

I have only a little background in logic, so I don't feel comfortable changing this part of the page, but I think it should be reverted to a form more compatible to the name "axiom of choice," if only for the fact that it's more self-contained.

Mo Anabre 20:11, 22 February 2007 (UTC)

The axioms here are taken verbatim from Kunen's book, where he calls the ninth axiom the axiom of choice. I have presented an argument higher on this talk page for keeping the entire set of axioms from one book; see the section Switch to Kunen's axioms.
I still think that it would benefit the article more to explain the various ways in which the axioms are not canonical than it would to pick a different noncanonical choice of axioms. CMummert · talk 21:06, 22 February 2007 (UTC)
I am aware of the discussion and your reasons for replacing it, but it begs several questions, some of which I've already proposed.
(1) If we're to decide on the stability aspect, we probably should decide based on the form most often used. In this way, the well-ordering theorem, while ZF-logically equivalent to the form I've seen used much more often, is not the same thing. This is my point with adding Zorn's lemma as an alternative. If we want to use any ZF-equivalent form and you want it to be as far from the canonical form as possible, let's just change it: list the axiom of choice as "Every non-empty partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element."
(2) The aspect of being self-contained. The axioms shown here should be self-contained (aside from the obvious necessity of defining metalogic/syntax). This specific argument is really what I'm focusing on the most. In my mind, the reason why I refuse to accept the well-ordering theorem as the axiom of choice is that it isn't self-contained. One must access another definition much more advanced than metalogic to find out what "is minimal under R" means.
Personally, if you just wanted references, I'll give you some references. Meanwhile, it seems that making a section on Kunen's axioms (and labelling as such) would be more within the bounds of reason.
Mo Anabre 19:13, 23 February 2007 (UTC)
If you would like to change the article so it uses the ZFC axioms from a different book, then please feel free to fix it; I have no attachment to Kunen's axioms, but I don't see how any other set of axioms is more "canonical" so I have no desire to change the article myself. I explained higher on the talk page why I think all 9 should come from the same book. CMummert · talk 19:48, 23 February 2007 (UTC)

Dash in page title

I moved "Zermelo–Fraenkel set theory" to "Zermelo-Fraenkel set theory" based on Wikipedia:Manual of Style (dashes). I was not aware that there was a discussion of changing the guideline on Wikipedia_talk:Manual_of_Style_(dashes)#Proposed_expanded_version_of_advice_on_hyphens_and_dashes_in_MoS.--Patrick 09:32, 9 June 2007 (UTC)

As a general rule, one should not move an important article like this without first discussing it on the talk page and getting a consensus. I have seen many articles moved the other direction, from hyphens to dashes. So you appear to be going against the trend. JRSpriggs 10:16, 9 June 2007 (UTC)
Weren't those moves in the other direction mostly done by the banned user Jon Awbrey, and weren't they also done without consensus? (That is certainly the case for this article, at least.) --Zundark 11:51, 9 June 2007 (UTC)
There is a general rule that names like this should use endashes. This is one case where JA got it right. The idea is that you use endashes to separate surnames of different persons, but hyphens where it's what Charles Matthews calls a "double-barreled name" of a single person (so Burali-Forti paradox rather than Burali–Forti paradox, because it's named after Cesare Burali-Forti rather than a Burali and a Forti. --Trovatore 23:25, 9 June 2007 (UTC)
Some were moved by Trovatore (talk · contribs) or Blotwell (talk · contribs). JRSpriggs 09:12, 10 June 2007 (UTC)
I don't mind moving it back if that is desired. But currently Wikipedia:Manual of Style (dashes) says that hyphens are preferred.--Patrick 10:46, 10 June 2007 (UTC)
There is an ongoing discussion at that MOS page about several changes to the style guide, including alowing en dashes in titles again. It might be bset to wait for that to resolve. — Carl (CBM · talk) 11:48, 10 June 2007 (UTC)

Now that the policy has been updated, how about we move it back to the correct title with en dash? Dicklyon 05:58, 15 June 2007 (UTC)

Heck, I just did it. RossBot will take care of the double redirects shortly, I bet. Dicklyon 06:01, 15 June 2007 (UTC)

Switch to Shoenfield's axioms: any objections?

Since many of Kunen's axioms are redundant, I propose to use Shoenfield's axioms instead, Shoenfield being a standard book on the subject as well, and probably the most accurate. They consist of: extensionality, regularity, union, power set, infinity, and replacement. (Shoenfield actually uses the "subset" and "replacement" schemas instead of the "union" axiom and "replacement" schema in the main text, but the equivalence of the two sets of axioms is mentioned in the exercises, the latter being more common.) I will change them if there are no objections. —The preceding unsigned comment was added by Neithan Agarwaen (talkcontribs) 16:37, 13 July 2007 (UTC)

Actually, I do kind of object to that. Redundancy, as I see it, is not really a problem. Having aussonderung and replacement listed separately is convenient for explaining the difference between Z and ZF, and I think it's more standard and more historical. It's not a huge deal, but on balance I prefer the status quo. --Trovatore 19:50, 13 July 2007 (UTC)
I'm not sure what "most accurate" means; I know of no serious objections to the overall correctness of Kunen or Jech's books on set theory. The "redundancy" is not a serious issue in my opinion. On one hand, as Trovatore points out, the current axioms are quite historically motivated and well known. Another issue, perhaps less serious, is that it is common to look at structures that don't satisfy replacement or power set, in which case the above list of axioms is no longer obviously enough to prove even pairing. I would have no objection to switching to Jech, although I see no advantage; but I think it would be a disservice to our readers to select an even more idiosyncratic selection of axioms, that will disagree with the majority of undergraduate texts, just to avoid some redundancy. — Carl (CBM · talk) 22:01, 13 July 2007 (UTC)
I agree with Carl. JRSpriggs 01:52, 14 July 2007 (UTC)
Well, I think there are enough objections already. Although I have to say, I do not really understand them. The axioms of Schoenfield are the same than Kunen's, without "set existence", "pairing", and "specification", and these can (and should) be mentioned as being derivable from the other axioms. In fact, I just mentioned Schoenfield so as to provide a standard reference for the axioms ("most accurate" in the treatment of set theory itself, I meant, not as far as the axioms are concerned), as has been asked elsewhere. So no information or clarity would be lost. On the other hand, having a minimal set of axioms not only makes the theory seem more elementary, but it is also greatly useful for the working mathematician, for the same reasons as it is useful to have less primitive symbols and less logical axioms/inference rules in formal systems. Also, Kunen's axiom of choice is quite nonstandard, I think, and the usual axiom "there exists a choice function on any set" is more common. Anyway, I agree that it is not really important. -- Neithan Agarwaen 11:27, 14 July 2007 (UTC)
There are some comments immediately following the last axiom which include identifying the axiom of pairing as redundant. If you want to add to those comments or emphasis them more, that would be helpful, and I hope it would satisfy your concerns. JRSpriggs 03:05, 15 July 2007 (UTC)

Z notation link

Is there any reason to keep the Z notation link and the corresponding category? Z notation is not really related to math. —Preceding unsigned comment added by 75.62.4.229 (talk) 07:33, 22 November 2007 (UTC)

"Union" textual vs. equation

For any set x, there is a set y such that the elements of y are precisely the members of the members of x.

The text says (all) the members of (all) the members. However, the equation says '', that is 'one member'. Is this correct ? I'm tagging the article as contradictory. --Hdante 08:13, 5 March 2006 (UTC)

The axiom of union is here correctly formulated. The axiom defines B := ∪A. B is called the union set of A. B collects precisely all sets C which are member of any set D in A. Otto ter Haar 12:02, 5 March 2006 (UTC)

That's true ! :-) --Hdante 17:26, 5 March 2006 (UTC)
The current formula doesn't say that, though. That formula says that B contains all the sets which are members of sets in A, but it doesn't say it contains only those sets. The formulation given in Axiom_of_union avoids this problem. Rsmoore (talk) 16:22, 17 March 2008 (UTC)
Both formulations are common; there's no problem. The formulation in the axiom of union article follows from the one here and a comprehension axiom. The axioms here are, for the sake of consistency, all taken from a single source (Kunen's book). The text here does not claim that the set A is literally the union, only that it contains all the elements that are in the union. As Kunen points out, this formulation is easier to work with when the goal is to verify that the axiom is satisfied by a particular model. — Carl (CBM · talk) 17:52, 17 March 2008 (UTC)

replacement implies comprehension

The article used to say that the ZFC axioms minus comprehension imply comprehension. Here is one proof that this is true when the empty set is assumed to exist. Let A be any set and assume we want to form the set . Let be any set in A such that (if there is no such set Z then W is the empty set). Let say that either and or and . Thus defines a function from A to A such that any element satisfying maps to itself, and all the other elements map to . Then the range of is exactly the set that we are trying to construct, and this exists by replacement on the formula . This proof doesn't need to be in the article, but I think the statement is interesting. CMummert 23:53, 4 September 2006 (UTC)

The problem is that you are assuming a version of the axiom of replacement which is not the one in the article. The one in the article says:
Notice that there is no limit on how many extra elements one can put into Y and thus it could be that Y might just be the same as A in your example. One needs the axiom of separation to convert this version of replacement into the one which you are assuming. JRSpriggs 08:20, 5 September 2006 (UTC)

While I expect the version stated in the article to be equivalent to the one CM is thinking of, there remains a problem. The description of replacement still states that the generated set Y is the co-domain of the function, which is no longer true. Correct me if I'm wrong, but the cryptic mention of a restriction to avoid paradox can't possibly account for this discrepancy since making the set larger risks creating more paradoxes, if anything.

I rephrased the English description to make it agree with the formal axiom. The cryptic comment about a restriction is referring to the restriction on the free variables of the formula. I don't think the two versions of replacement are equivalent in the absence of comprehension, which is why JRSpriggs's comment settled the matter for me. CMummert · talk 23:52, 27 January 2007 (UTC)
Let me expand on what CMummert said about the restriction on free variables. Y should not appear free in φ. If it did, then the existential quantification over it on the right side of the implication would cause trouble because then the function implicitly defined by φ would change from what it was on the left. JRSpriggs 12:40, 28 January 2007 (UTC)

Huh?

What is this?:

Why is it possible that and in the same context? The formula spans over z twice separately, so we cannot protest too much here, but x seems to be the same kind of set, so the first z spans over members of x, which then must be a set, but the second z spans over sets of sets. Is this intended to mean anything, or is it a typo where should be replaced by ? ... said: Rursus (bork²) 13:21, 25 February 2009 (UTC)

Oh, I forgot: in the extension of axiom 1. ... said: Rursus (bork²) 13:26, 25 February 2009 (UTC)
OK, I got it wrong. It is quite possible that x, y and z are sets of sets, which seems to be an assumption in this set theory. ... said: Rursus (bork²) 13:39, 25 February 2009 (UTC)
As the lead says "all individuals in the universe of discourse are such sets". Each thing is a set and all of its elements are also such sets. Thus all the variables and constants in this theory represent sets of sets of sets of sets of ... ad infinitum. If you keep moving down from a set to one of its elements (also such a set), then eventually you reach the empty set which has no element to choose. See pure set. JRSpriggs (talk) 22:26, 25 February 2009 (UTC)

It seems like it would be easy to use two different variables instead of writing z twice.

metamath

I removed this paragraph:

One piece of evidence bearing on ZFC as a foundation of mathematics is Metamath, an ongoing web-based project that seeks to derive much of contemporary mathematics from the ZFC axioms, first order logic, and a host of definitions, with all proofs verified by machine. As of early 2008, the Metamath database includes about 8000 proved theorems. This project can be seen as being in the same spirit as Bertrand Russell's Principia Mathematica, except grounded in logical and nonlogical axioms that benefit from nearly a century of subsequent research.

There's maybe some better way to mention Metamath in the article, but I find the claims in the paragraph above to be a bit overstated. 75.62.6.87 (talk) 20:27, 5 April 2009 (UTC)

σ set theory

I removed a link to the arxiv about an alternative set theory. Because almost anyone can publish almost anything on the arxiv, I don't think we should generally be using preprints there as references. If some published text or journal refers to the theory, that would make me want to consider whether to include it. Are there any references like that? Really I am interested in evidence thqat set theorists other than the author are interested in the concept. — Carl (CBM · talk) 00:42, 16 July 2009 (UTC)

Some problems and some questions

I am finding it difficult to make sense of the WikiPedia specification of ZFC. Below are some of the problems and questions I have. Can some kind reader please help me understand what I have misunderstood. W J Eckerslyke (talk) 12:51, 1 February 2009 (UTC)

Chickens and eggs

ZFC “is the standard form of axiomatic set theory and as such is the most common foundation of mathematics”. Yet its “universe of discourse” is claimed to comprise “all mathematical objects”. How can this be? How can ZFC claim to be based on objects which are supposedly based on ZFC? At the very least some clarification is needed.

What clarification would you propose? — Carl (CBM · talk) 20:02, 1 February 2009 (UTC)
As far as I know, any mathematical object can be modeled in set theory. That is, it can be represented by a set and it properties and relationships can also be represented by sets.
As far as I know, no claim is made that ZFC is based on anything other than logic and sets; and no claim is made that other mathematical objects are necessarily based on ZFC or sets. JRSpriggs (talk) 10:53, 2 February 2009 (UTC)

What about non-mathematical objects?

Why should a set theory be confined to discussion of mathematical objects only? Surely a set theory, even an axiomatic one, could usefully be applied to sets of books, stars, or badgers?

This is true from a certain point of view, but for the mathematical study of set theory, there is no need for sets that contain non-mathematical objects, and indeed no need for set that contain anything other than sets. The effects of including urelements in set theory are well known and generally not of much mathematical interest. — Carl (CBM · talk) 20:04, 1 February 2009 (UTC)
As Carl says, non-mathematical objects could be included by making them urelements. This result in ZFU (Zermelo–Fraenkel set theory with urelements). But ZFU itself can be modeled in ZFC and is not especially interesting to a mathematician; it just introduces fruitless complexity. JRSpriggs (talk) 10:59, 2 February 2009 (UTC)

Individuals

Is it really necessary, or even desirable, to assume that “all individuals ... are sets”? This assumption certainly offers some potential simplification, but it also causes a serious problem which ZFC does not seem to address. The question is whether for all individuals x it is true that {x}=x. If this is not true then all individuals have the same members, i.e. none, and are thus indistinguishable from each other and from the empty set. But if it is true then neither the Axiom of Regularity nor the Axiom of Infinity bears examination!

In set theory with urelements, the axiom of extensionality has to be changed for exactly the reason you describe. This is describe in Jech's book, for example. But since ZFC has no urelements this is not an issue with ZFC. So I don't see how this relates to the present article, which is about ZFC in particular, not set theory in general. — Carl (CBM · talk) 20:10, 1 February 2009 (UTC)

Domain of discourse

Are there any individuals (other than the empty set)? “Many authors require a nonempty domain of discourse as part of the semantics of the first-order logic in which ZFC is formalized.” I cannot see why this should be optional. If there is no given 'domain of discourse' then the whole thing just enables the construction of confections of the empty set.

A very few authors use free logic, but most of them require that a domain of discourse must be nonempty, so they agree with you. But what relevance does this have to the article? — Carl (CBM · talk) 20:08, 1 February 2009 (UTC)
I support "free logic" which is actually merely valid logic. "Logics" which assume the existence of an individual in the domain of discourse are not valid, i.e. not real logic. JRSpriggs (talk) 11:05, 2 February 2009 (UTC)

Active sets?

“The axioms of ZFC govern how sets behave and interact.” Strange wording! Sets are essentially passive objects, and do not behave or interact at all. It seems to me the axioms do no more than specify what sets are deemed to “exist”.

In ordinary English, "behave" and "interact" suggest change and thus the passage of time, which does not apply to ZFC. However, there certainly are relationships between sets which are definable in terms of the element relation. Typically, one thinks of stages in the process of definition as being constructions which occur over time in our minds. So we apply words which, perhaps, are describing our mental state more than the actual mathematical objects. Timelessness is a property of the content of mathematics generally, not just set theory; but mathematicians live in time. JRSpriggs (talk) 11:14, 2 February 2009 (UTC)

Consistency

There seems to be some doubt that ZFC is in fact consistent. Doubtless there is no actual proof of its consistency, but at first glance there really does not seem to be enough in it to allow the possibility of inconsistency. However, its blithe use of reference to “any property” (Axiom 3) and to “any formula” (Axiom 6), without any attempt to limit the scope of the “any”, clearly leaves infinite scope for things to go wrong, and consistency is therefore likely to depend on just what scope is chosen for properties and formulas. It also seems to be left open whether or not “the background logic includes equality” (Axiom 3). The conclusion might perhaps be that ZFC may or may not have any consistent realisations, but it is very likely to have some inconsistent ones.

There is little doubt among mathematicians that ZFC is consistent. In particular, there is no great suspicion that it is actually possible to prove a contradiction from the ZFC axioms. On the other hand (and this point of view dates back to Zermelo), if ZFC is inconsistent, the most likely way to discover that is to rigorously study its conssequences.
Some mathematicians have other objections to ZFC – they might dislike classical logic, or not like infinite structures. But the percentage of such mathematicians in the general population is vanishingly small. — Carl (CBM · talk) 20:07, 1 February 2009 (UTC)

Infinite sets

“The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω.” Indeed, it would seem that ω is the only set whose existence this axiom guarantees, even if there is a non-empty domain of discourse. So perhaps this should be called the Axiom of ω?

The common name for it is the axiom of infinity, and this is the name that all set theory texts I have seen use. The axiom is so named because it directly postulates the existence of an infinite set. In some statements of the axiom it does not literally say that ω exists, only that some set containing ω exists. — Carl (CBM · talk) 19:57, 1 February 2009 (UTC)
Assuming you mean a set containing ω as a subset rather than as an element, you've taken the words out of my mouth. The set could be any limit ordinal, or it could be something like ω union the successor sequence of {{{∅}}}. The obvious way to deduce that ω is a set would be to apply the axiom of specification, except that I can't at the moment see a predicate we could use that is within the ZF axioms. -- Smjg (talk) 12:55, 5 June 2010 (UTC)
Sometimes the axiom of infinity is stated as ``There exists an infinite set" (with careful meaning given to infinite). Then we define ω to be the intersection of all infinite sets. (Note that this intersection exists only if there is at least one infinte set. Otherwise we get nonsense). Setitup (talk) 22:34, 2 March 2010 (UTC)
The intersection of all infinite sets is empty. For example, consider the infinite set consisting of just the even natural numbers, and the infinite set consisting of just the odd natural numbers. One can define ω as the intersection of all infinite sets that contain the empty set and are closed under the map . — Carl (CBM · talk) 02:38, 3 March 2010 (UTC)
But the axioms don't enable us to
  • know what is meant by an infinite set (that said, that condition is redundant)
  • conceive the infinite class of sets having the property and identify the intersection of them all. -- Smjg (talk) 12:55, 5 June 2010 (UTC)
Not clear what you mean by any of this. The intersection of all sets satisfying some predicate P is completely well-specified, given that at least one set satisfies P. And the predicate "is an infinite set" is certainly definable in the language of set theory. Carl is certainly correct that the intersection of all infinite sets is empty.
Your remarks about the axioms "not enabling us" to know or conceive of these things are correct, but maybe not for the reason you think. Axioms don't "allow us to know or conceive of" anything; that's not the function of axioms. Oh, sure, the axiomatic method can derive new knowledge from old knowledge, and such knowledge can be conceptualized. But that wasn't what you seemed to be talking about. --Trovatore (talk) 20:16, 5 June 2010 (UTC)
To Smjg: See axiom of infinity#Extracting the natural numbers from the infinite set for two different ways of extracting ω from a superset of it.
An infinite set is a set which is not a finite set. See the section finite set#Necessary and sufficient conditions for finiteness for a variety of ways of characterizing finiteness. JRSpriggs (talk) 17:03, 6 June 2010 (UTC)
Basically, what I meant is that ZF merely postulates the existence of one set having this property. "The intersection of all sets satisfying some predicate P" is well-specified only if "all sets satisfying some predicate P" is. This, in turn, is well-specified in a set theory that includes the axiom of unrestricted comprehension, but ZF deliberately doesn't include it.
But now I see: a natural number is a set:
  • where every element is either 0 (∅) or the successor of a sibling element (x ∪ {x})
  • and which itself is either 0 (∅) or the successor of one of its elements (x ∪ {x})
I was trying to do something like that, but on the infinite set as a whole instead of each of its elements. Thanks, that's cleared it up. -- Smjg (talk) 01:14, 10 June 2010 (UTC)
Well, good, but you're still missing a point. You don't need unrestricted comprehension to make the intersection well-specified. The intersection is just the collection of all sets x such that x is in every set y for which P(y) holds. That works just fine to specify the intersection, without no need to collect all such y into a completed totality. --Trovatore (talk) 01:19, 10 June 2010 (UTC)
Oh, maybe you're concerned that there might be no set collecting all such x. But there is, granted that there's at least one y0 such that P(y0) holds, because then the desired collection is a subset of y0. --Trovatore (talk) 01:21, 10 June 2010 (UTC)
So essentially, you're exercising ZF's axiom of specification on the postulated infinite set with the predicate Q(x) "every set containing ∅ and the successor of every one of its elements contains x". Such that you don't need to worry about collecting all y satisfying P(y), only about whether P(y) ⇒ xy. Have I got that right? -- Smjg (talk) 23:56, 11 June 2010 (UTC)

Power set

There is what seems to be a dangerous imprecision in the presentation of the axiom of power set. The text says “The power set of x is the class whose members are every possible subset of x.” This “possible” may be interpreted as meaning “conceivable”, whatever that means. But that would be wrong. In this context “possible” can only mean “possibly existing in accordance with these axioms”. This is perhaps clearer in the formal definition: in which the scope of the is more obviously limited to sets allowed under ZFC.

I've reworded this to avoid the word "possible", as this word doesn't seem helpful here, and is potentially confusing. --Zundark (talk) 13:56, 1 February 2009 (UTC)
Thanks for response. However, I had and have no objection to the word "possible", but merely sought clarification of its meaning. Which I still seek. Are you able to confirm that the ZFC power set includes only subsets allowable under the other ZFC axioms? W J Eckerslyke (talk) 17:51, 1 February 2009 (UTC)
I removed the word "possible" because I had an objection to it. As for your question: intuitively, it contains all subsets (formally - see the formula). --Zundark (talk) 19:25, 1 February 2009 (UTC)
The informal goal of the ZFC axioms is to describe statements that are true about the collection of all sets. So the axiom of power set is intended to say that for any set X there is another set containing all subsets of X. I don't think that the statement "only subsets allowable under the other ZFC axioms" actually makes sense if you examine it more closely – because the collection of all sets satisfies each the axioms of ZFC, it's not at all clear that any set is proscribed by the ZFC axioms. — Carl (CBM · talk) 19:59, 1 February 2009 (UTC)
I am not sure it is entirely true that no set is proscribed by the ZFC axioms: the Axiom of Regularity seems to put some (reasonable) limits on what sets may be deemed to exist. But, apart from that, ZFC is indeed short of proscriptions. However, the axioms do prescribe a minimum population of sets which the ZFC adherent is required to believe in, starting with the empty set and the von Neumann ordinal, and including all sets whose existence is guaranteed by virtue of e.g. the Axioms of specification, pairing, union and replacement. You seem to be saying that in addition to this guaranteed minimum population the adherent is allowed to believe in any other conceivable sets which are consistent with the axioms. If that is so then I think it should be made clear in the article. But even then these additional sets must surely be optional, and anything provable under ZFC must surely not be allowed to assume more than the guaranteed minimum population of sets. Nor, I believe, can the Axiom of power set be deemed to guarantee more subsets than are required by the other axioms. Which was my original point. W J Eckerslyke (talk) 10:41, 3 February 2009 (UTC)
It's true that the axiom of regularity restricts the universe of ZFC to well-founded sets instead of arbitrary sets. Kunen discusses this choice at some length in his book.
The minimum model of ZFC is the constructible universe, and thus it is true in some sense that these would be all the sets that one "has to believe in". I'm not sure whether this point really warrants a long discussion in the article, though. In particular, the standard way of describing the power set axiom is that it guarantees that for any set X there is another set Y containing all subsets of X. If you relativize this to a model M you get: for every X there is a Y which contains every subset of X that is in M. But the intended model is the collection of all (well-founded) sets, in which case Y does include all subsets of X. — Carl (CBM · talk) 14:14, 3 February 2009 (UTC)
I added a section on the cumulative hierarchy, which really was a glaring omission. But I don't know exactly what other changes you are proposing. — Carl (CBM · talk) 16:07, 3 February 2009 (UTC)
Let us call a set "founded", if the transitive closure of the singleton of the set is well-founded with respect to the element relation. If a set is founded, then all its elements are founded. If all elements of a set are founded, then the set is founded. If a set is founded, then its powerset is founded. None of the operations for creating sets allowed by the axioms of ZFC can create anything unfounded if applied to things which are founded. JRSpriggs (talk) 03:11, 4 February 2009 (UTC)
I'm just an ignoramus trying to make sense of all this, which I have not yet succeeded in doing, despite the kind assistance of several contributors, for which I am very grateful. I note that "the minimum model of ZFC is the constructible universe", and I assume that this means that "the constructible universe" is the most that anybody proving theorems on the basis of ZFC can rely on. If so, that greatly limits what can be proved on the basis of ZFC, and if true this is a point which would be of sufficient importance to warrant a brief mention in the article, IMHO. The fact that "the intended model" is somewhat larger does not seem to alter this. Moreover, if to enable this larger potential model we allow a set to be anything pure and well-founded that I can imagine then I don't see why we need Axioms #3 to #8 at all: I can perfectly well imagine the existence of such sets without being given explicit licence to do so. W J Eckerslyke (talk) 16:49, 4 February 2009 (UTC)
The constructible universe is the smallest model of ZFC containing all the ordinals. But this does not mean that it is possible to prove in ZFC that every set is constructible, and so one cannot particularly "rely on" constructibility when proving things in ZFC. By analogy, the only elements of a field of characteristic 0 that must exist are the rational numbers (in this context the rationals are the prime field of characteristic 0). However, one cannot assume when proving something about a field of characteristic 0 that every element is rational, and indeed the field of rationals is not a very interesting field of characteristic 0. Similarly, although L satisfies the ZFC axioms, it is not one of the more interesting models of ZFC to set theorists.
Re "I can perfectly well imagine the existence of such sets without being given explicit licence to do so." – keep in mind the axiomatic method. The goal is to isolate a few particular axioms about sets, each of which is intuitively justified, and from which it is possible to prove many interesting theorems. The intuitive justification for the ZFC axioms is the cumulative hierarchy, and the ZFC axioms are strong enough to prove essentially all theorems of non-set-theoretic mathematics. In this sense the axioms of ZFC represent a particularly successful application of the axiomatic method, showing that no further intuitions about the nature of sets, beyond those conveyed by the ZFC axioms, are required for the bulk of mathematics. — Carl (CBM · talk) 18:16, 4 February 2009 (UTC)

Definability

“Much research has sought to characterize the definability (or lack thereof) of certain sets whose existence AC asserts.” Is that a good use of effort? Would it not be more sensible instead to develop a set theory based on a set definition capability, and to accept the existence of all sets which can be defined by it and no others?

It might seem more natural to you, but set theorists have found that the study of the relationshup between definability and the axiom of choice to be very fruitful. In particular, one reason that the constructible universe has many of its properties is that it has definable choice functions for every nonempty set. On the other hand, topological regularity properties such as projective determinacy require that there is no definable choice function on the reals of a certain complexity. — Carl (CBM · talk) 20:01, 1 February 2009 (UTC)