73
votes

Accepted

### How should a "working mathematician" think about sets? (ZFC, category theory, urelements)

Set theory provides a foundation for mathematics in roughly the same way that Turing machines provide a foundation for computer science. A computer program written in Java or assembly language isn't ...

43
votes

### Does anyone still seriously doubt the consistency of $ZFC$?

When I was a graduate student Jack Silver was famous for trying to refute first, the existence of measurable cardinals, and second, the consistency of ZFC. His attempt at refuting measurable cardinals ...

Community wiki

34
votes

### Does anyone still seriously doubt the consistency of $ZFC$?

For decades I was not particularly suspicious about the consistency of ZFC but I was rather surprised about how it had become the standard choice when it contained axioms such as Replacement, which ...

Community wiki

27
votes

### How should a "working mathematician" think about sets? (ZFC, category theory, urelements)

I prefer to think of ZFC as a proposed model of mathematics. I want to emphasize both words "proposed" and "model". For comparison, consider quantum mechanics. It can be modeled — as far as we know, ...

26
votes

### Does anyone still seriously doubt the consistency of $ZFC$?

We have enough experience with ZFC that I think we can say it is consistent with some confidence.
This isn't too surprising --- I don't know if there's a way to say this precisely, but morally ...

Community wiki

26
votes

Accepted

### Hilbert's sixth problem and QFT description

The reason is that there is no mathematically rigorous construction of any interacting quantum field theory in four space-time dimensions to this date. Because of that, one has not been able so far to ...

23
votes

Accepted

### What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?

There's a certain confusion underlying your question, which Andreas Blass's answer is trying to point out. Let me see if I can explain it in different words.
You say, “the negation of Con(ZFC) proves ...

23
votes

### Was there a time in mathematics when a counterexample was wrong?

A very famous and important example of a counterexample that was found to be defective occurred in set theory. As Georg Cantor developed the theory of infinite sets, he proved that some infinite sets ...

23
votes

### Does anyone still seriously doubt the consistency of $ZFC$?

I interpret the question factually: Do there exist professional mathematicians who seriously doubt the consistency of ZFC? The answer is yes. Here are two examples (though sadly, both mathematicians ...

Community wiki

22
votes

### How should a "working mathematician" think about sets? (ZFC, category theory, urelements)

There are already some excellent answers explaining in what senses ZFC can still be a foundation for most mathematics. But it also seems appropriate to mention some ways in which ZFC is insufficient ...

22
votes

Accepted

### When does a topos satisfy the axiom of regularity?

The relationship between toposes and set theories was studied comprehensively in
Steve Awodey, Carsten Butz, Alex Simpson, Thomas Streicher: Relating first-order set theories, toposes and categories ...

19
votes

### What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?

$\DeclareMathOperator\BB{BB}$Philosophical issues, like acceptance (or non-acceptance) of large cardinals, won't affect $\BB(n)$, because the busy beaver function is defined arithmetically and so ...

17
votes

### Does anyone still seriously doubt the consistency of $ZFC$?

George Boolos, eminent philosopher and logician, wrote as follows (perhaps slightly tongue-in-cheek):
He continued for a bit, and then...
(From G. Boolos, Logic, Logic, and Logic, Chapter 8: "...

Community wiki

13
votes

Accepted

### Logical completeness of Hilbert system of axioms

The original is Alfred Tarski's book "The completeness of elementary algebra and geometry", which was due to appear in 1940 but never made it into print because of the outbreak of WW2. An edition ...

13
votes

Accepted

### Applications of ZFA-Set Theory

Although you prefer models other than permutation models, let me point out an appearance of permutation models, particularly the basic Fraenkel model, at the border between computer science and logic. ...

13
votes

### Axioms for the category of groups

This is not really in the spirit of the examples you give but it is at least a set of purely categorical properties.
Proposition: A category $C$ is the category of models of a Lawvere theory iff it ...

13
votes

### Does anyone still seriously doubt the consistency of $ZFC$?

This is basically a long comment. I like Nik Weaver's careful distinction between the question of whether ZFC is consistent and the question of whether it is (e.g. arithmetically) sound, and would ...

Community wiki

12
votes

Accepted

### Does bounded Zermelo construct any cumulative hierarchy?

KP (Kripke-Platek set theory) is the most well-known fragment of $\sf{ZF}$ which suffices for the development of the rank function, thus $\sf{KPR}$ = $\sf{KP}$ + "for all ordinals $\alpha$, $V(\...

12
votes

Accepted

### Is this set theory equivalent to ZFC?

Accepting the convention that it is a logical axiom that the universe is nonempty, the answer is yes. We will formalize the transitive closure axiom schema (TC) as follows: for any definable (with ...

12
votes

Accepted

### Is every set being cardinal definable consistent with ZF + negation of Choice?

This is consistent. Kanovei constructed a model $M$ with an infinite Dedekind finite set of reals which is lightface projectively definable. By descending to $L(R),$ we can further assume it satisfies ...

11
votes

Accepted

### Does the axiom schema of Replacement follow from the abstract notion of the iterative conception of sets?

In my blog post Transfinite recursion as a fundamental principle in set theory, I prove that the principle of transfinite recursion is equivalent to the replacement axiom.

11
votes

Accepted

### Is full Replacement provable in Z + Ordinal Replacement?

No. Consider the model $\langle V_{\omega_1},\in\rangle$. This is a model of Zermelo's theory, but all ordinals in it are countable. Thus, it satisfies the ordinal-replacement axiom, since the image ...

11
votes

### Is axiom of constructibility $V = L$ consistent with Tarski–Grothendieck set theory?

(I made my answer community wiki because the previous comments cover the important points of the answer.)
You can find the relationship between inaccessible cardinals and Grothendieck universes on ...

Community wiki

11
votes

Accepted

### Is choice over definable sets equivalent to AC over axioms of ZF-Reg.?

Yes, indeed this kind of choice in general doesn't imply $\mathsf{AC}$ over $\mathsf{ZF}-\mathsf{Reg}$.
I will reason in $\mathsf{ZFC}$ and construct an interpretation of $\mathsf{ZF}-\mathsf{Reg}$, ...

10
votes

Accepted

### Can rules of set theory be founded by paralleling parts of atomic Mereology?

Many people find it natural to consider the set-theoretic analogue of mereology to be the inclusion relation $\subseteq$ rather than $\in$, since after all, $\subseteq$ is reflexive and transitive ...

10
votes

Accepted

### Axioms for the category of groups

As requested, here is an answer summarizing axioms for the category of groups that were given by Pierre Leroux, and which I learned from an MSE answer of Arnaud D. The category of groups is the unique ...

10
votes

Accepted

### How short can the axioms of propositional logic be?

No. Take the set of truth values to be $\{\top, \bot, P, \neg P, Q, \neg Q\}$, with $\neg$ defined in the obvious way and $\to$ defined by cases: $p \to \top$ and $\bot \to p$ are both $\top$, $\top \...

9
votes

Accepted

### Can you formulate a theory stating that a truth predicate does not exist for first order set theory?

The assertion that there is (or is not) a truth predicate is expressible in the second-order language of set theory, but assuming consistency, not by any first-order assertion.
Second-order. In the ...

9
votes

Accepted

### What would be the effect of replacing Separation by Injective Replacement?

Replacing separation with injective replacement is equivalent to adding the usual replacement axiom, and so your theory $Z^{inj}$ is the same as ZF. From this, you can easily answer all your questions....

9
votes

### How should a "working mathematician" think about sets? (ZFC, category theory, urelements)

This complements the other answers. Let's take the natural numbers as an example for discussion.
When we use mathematics, we typically want to use properties of the natural numbers: after every ...

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