**48**

votes

**2**answers

14k views

### Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...

**23**

votes

**2**answers

2k views

### Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ...

**117**

votes

**15**answers

21k views

### Why worry about the axiom of choice?

As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't ...

**67**

votes

**9**answers

19k views

### solving f(f(x))=g(x)

This question is of course inspired by the question How to solve f(f(x))=cosx
and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...

**34**

votes

**7**answers

6k views

### Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...

**34**

votes

**6**answers

2k views

### Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?

If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The ...

**7**

votes

**2**answers

909 views

### How to construct a continuous finite additive measure on the natural numbers

I want to find some condition to construct a continuous finitely additive measure on the natural numbers, i.e. $f:P(\mathbb{N})\rightarrow [0,1]$ such that $f(\{n\})=0$, and $f$ is an additive ...

**11**

votes

**1**answer

605 views

### Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?

Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense ...

**70**

votes

**3**answers

7k views

### Does every non-empty set admit a group structure (in ZF)?

It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary ...

**13**

votes

**8**answers

3k views

### Forcing as a tool to prove theorems

It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to ...

**28**

votes

**0**answers

1k views

### Concerning the various proofs from the axiom of choice that R^3 admits of surprising geometrical decompositions into circles, skew lines and so on: can we prove in any instance that there are no Borel such decompositions? Or that AC is required?

This question follows up on a comment I made on Joseph O'Rourke's
recent question, one of several questions here on mathoverflow
concerning surprising geometric partitions of space using the axiom
of ...

**83**

votes

**9**answers

13k views

### Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH; How far wrong could the Continuum Hypothesis be?
...

**50**

votes

**15**answers

13k views

### Most 'unintuitive' application of the Axiom of Choice?

It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis. Even ...

**52**

votes

**2**answers

4k views

### Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...

**35**

votes

**4**answers

9k views

### Non Borel sets without axiom of choice

This is a simple doubt of mine about the basics of measure theory, which should be easy for the logicians to answer. The example I know of non Borel sets would be a Hamel basis, which needs axiom of ...

**39**

votes

**7**answers

2k views

### How would one even begin to try to prove that a simple number-theoretic statement is undecidable?

This question is closely related to this one: Knuth's intuition that Goldbach might be unprovable. It stems from my ignorance about non-standard models of arithmetic. In a comment on the other ...

**27**

votes

**5**answers

4k views

### Completion of ZFC

I attended a talk given by W. Hugh Woodin regarding the Ultimate L axiom and I wanted to verify my current understanding of what the search for this axiom means. I find it to be a fascinating topic ...

**17**

votes

**2**answers

3k views

### Large cardinal axioms and Grothendieck universes

A cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of ...

**28**

votes

**2**answers

2k views

### What interesting/nontrivial results in Algebraic geometry require the existence of universes?

Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (link). I'm wondering if there are any results in AG that ...

**19**

votes

**3**answers

2k views

### Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in ...

**33**

votes

**3**answers

939 views

### Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point.
Theorem. Suppose $P$ and $Q$ are posets ...

**28**

votes

**3**answers

2k views

### Even XOR Odd Infinities?

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$.
...

**19**

votes

**7**answers

1k views

### Does every set admit a rigid binary relation? (and how is this related to the Axiom of Choice?)

Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has no nontrivial automorphisms.
Under the Axiom of Choice, every set is ...

**9**

votes

**4**answers

3k views

### Pi1-sentence independent of ZF, ZF+Con(ZF), ZF+Con(ZF)+Con(ZF+Con(ZF)), etc.?

Let
ZF1 = ZF,
ZFk+1 = ZF + the assumption that ZF1,...,ZFk are consistent,
ZFω = ZF + the assumption that ZFk is consistent for every positive integer k,
... and similarly define ZFα ...

**6**

votes

**3**answers

870 views

### Tractability of forcing-invariant statements under large cardinals

It is usual to mention theorems of the kind:
Th. Assume there is a proper class of Woodin cardinals, $\mathbb{P} $ is a partial order and $G \subseteq \mathbb{P}$ is V-generic, then $V \models \phi ...

**14**

votes

**3**answers

615 views

### Singularizing forcing of “small” cardinality?

Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size ...

**14**

votes

**2**answers

1k views

### Does ZFC prove the universe is linearly orderable?

It is consistent with ZFC that the universe is well-ordered, e.g. in $V=L$ where global choice holds. I also know that it is consistent that global choice fails (although I have no immediate example ...

**16**

votes

**1**answer

1k views

### Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...

**9**

votes

**1**answer

558 views

### Resembling the Levy Collapse

Suppose $\kappa$ is a weakly compact cardinal. Is there a $\kappa$-c.c. forcing $\mathbb{P}$ such that $\mathbb{P} \subseteq V_\kappa$ and $\Vdash_{\mathbb{P}} \kappa = \aleph_1$, where $\mathbb{P}$ ...

**17**

votes

**2**answers

842 views

### If ZFC has a transitive model, does it have one of arbitrary size?

It is known that the consistency strength of $\sf ZFC+\rm Con(\sf ZFC)$ is greater than that of $\sf ZFC$ itself, but still weaker than asserting that $\sf ZFC$ has a transitive model. Let us denote ...

**10**

votes

**3**answers

871 views

### How elementary can we go?

It is a theorem of A. Levy, if $\kappa$ is an inaccessible cardinal, then $V_\kappa\prec_{\Sigma_1} V$ namely $V_\kappa$ is an elementary submodel when considering only $\Sigma_1$ sentences.
One ...

**6**

votes

**2**answers

659 views

### Is there one binary operation foundational for set theory?

The membership relationship "$\epsilon$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\epsilon$". Naturally, the question arises ...

**15**

votes

**4**answers

2k views

### Why is there no Borel function mapping every countable set of reals outside itself?

A choice function maps every set (in its domain) to an element of itself. This question concerns existence of an anti-choice function defined on the family of countable sets of reals. In an answer to ...

**10**

votes

**2**answers

276 views

### Sets that are not $\infty$-Borel

I have seen a few techinques for proving that certain sets of real numbers are $\infty$-Borel (definition) but it just occurred to me that I don't know of any way to prove that a set of real numbers ...

**5**

votes

**1**answer

326 views

### Infinite graphs isomorphic to their line graph

The only finite connected graphs $G$ that are isomorphic to their line graph $L(G)$ are the cycle graphs $C_n$ (see this link for example).
There are connected countable graphs that are isomorphic to ...

**14**

votes

**3**answers

446 views

### Does every set $X$ have a topology for which the only continuous self-surjection is the identity map?

This question is a special case of Dominic van der Zypen's question Reconstructing relations with the image relation of a topology, as discussed in the comments, particularly the comment of Eric ...

**2**

votes

**1**answer

458 views

### Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based
on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...

**137**

votes

**27**answers

17k views

### What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."
For example, suppose A is an abelian group such that every ...

**50**

votes

**5**answers

9k views

### Inaccessible cardinals and Andrew Wiles's proof

In a recent issue of New Scientist (16 Aug 2010), I was surprised to read that a part of Wiles' proof of Taniyama-Shimura conjecture relies on inaccessible cardinals.
Here's the link:
...

**54**

votes

**18**answers

6k views

### Proofs of the uncountability of the reals.

Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem. At first, I was excited to see a variant proof (as it did not use the diagonal ...

**41**

votes

**2**answers

4k views

### A question about ordinal definable real numbers

If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent
when the following statement is added to it as a new axiom?
"There exists a denumerably ...

**30**

votes

**13**answers

3k views

### Cardinalities larger than the continuum in areas besides set theory

It seems that in most theorems outside of set theory where the size of some set is used in the proof, there are three possibilities: either the set is finite, countably infinite, or uncountably ...

**23**

votes

**10**answers

43k views

### What practical applications does set theory have?

I am a non-mathematician. I'm reading up on set theory. It's fascinating, but I wonder if its found any 'real-world' applications yet. For instance, in high school and we were learning the properties ...

**16**

votes

**3**answers

2k views

### Is there a computable model of ZFC?

Background
Assuming ZFC is consistent, then by downward Löwenheim–Skolem, there is a countable model (M,$\in$) of ZFC. Since the universe M is countable, we may as well think of it as actually being ...

**31**

votes

**1**answer

2k views

### Dual Schroeder-Bernstein theorem

This question was motivated by the comments to Dual of Zorn's Lemma?
Let's denote by the Dual Schroeder-Bernstein theorem (DSB) the statement
For any sets $A$ and $B$, if there are ...

**27**

votes

**1**answer

1k views

### Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?

In comments on Quora (see, for example, here, here, here), Ron Maimon has repeatedly expressed the strong opinion that Hilbert's program was not killed by Gödel's results in the way typically ...

**15**

votes

**9**answers

2k views

### Examples of ZFC theorems proved via forcing

This is an old suggestion of Joel David Hamkins at the end of his answer to this question: Forcing as a tool to prove theorems
I just noticed it while trying to understand his answer. But indeed it ...

**14**

votes

**4**answers

3k views

### What would be some major consequences of the inconsistency of ZFC?

I was happily surfing the arXiv, when I was jolted by the following paper:
Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by ...

**23**

votes

**4**answers

2k views

### What axioms are used to prove Godel's Incompleteness Theorems?

I understand Godel's Incompleteness Theorems to be statements about effectively generated formal systems, which basically makes them theorems about algorithms. This is cool, because despite being ...

**10**

votes

**4**answers

3k views

### Finite axiom of choice: how do you prove it from just ZF?

The axiom of choice asserts the existence of a choice function for any family of sets F. Suppose, however, that F is finite, or even that F just has one set. Then how do we prove the existence of a ...