**91**

votes

**15**answers

17k 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 ...

**21**

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 ...

**60**

votes

**9**answers

16k 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 ...

**38**

votes

**2**answers

12k 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 ...

**11**

votes

**1**answer

426 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 ...

**31**

votes

**7**answers

5k 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 ...

**71**

votes

**9**answers

11k views

### Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? Completion of ZFC )
Background
The Continuum Hypothesis (CH) posed by Cantor in 1890 asserts that $ ...

**46**

votes

**2**answers

3k 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 ...

**30**

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 ...

**26**

votes

**0**answers

946 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 ...

**27**

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 $\exists x(Sx = 0)$.
...

**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 ...

**17**

votes

**2**answers

746 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 ...

**9**

votes

**1**answer

508 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 $\mathbb{P}$ ...

**6**

votes

**2**answers

580 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 ...

**10**

votes

**2**answers

240 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 ...

**120**

votes

**26**answers

14k 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 ...

**57**

votes

**3**answers

6k 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 ...

**40**

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 ...

**54**

votes

**16**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 ...

**29**

votes

**4**answers

7k 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 ...

**26**

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 ...

**15**

votes

**2**answers

2k 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 ...

**26**

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 ...

**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 ...

**27**

votes

**3**answers

792 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 ...

**27**

votes

**1**answer

844 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 ...

**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 ...

**8**

votes

**2**answers

822 views

### Is it possible for countably closed forcing to collapse $\aleph_2$ to $\aleph_1$ without collapsing the continuum?

Suppose the continuum is larger than $\aleph_2$. Does there exist a countably closed notion of forcing that collapses $\aleph_2$ to $\aleph_1$, but does not collapse the continuum to $\aleph_1$? ...

**22**

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 ...

**14**

votes

**2**answers

740 views

### Questions about $\aleph_1-$closed forcing notions

"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including:
1) $GCH$ fails ...

**17**

votes

**2**answers

1k views

### What is a Choice Principle, really?

This question is quite soft, and I apologize in advance if it borderline off-topic.
When working in theories between ZF and ZFC the term "choice principle" is heard quite often. For example:
$\quad$ ...

**11**

votes

**3**answers

981 views

### On statements independent of ZFC + V=L

Let $V=L$ denote the axiom of constructibility. Are there any interesting examples of set theoretic statements which are independent of $ZFC + V=L$? And how do we construct such independence proofs? ...

**13**

votes

**1**answer

402 views

### Unpublished works of Woodin on SCH and Radin forcing

There are many unpublished results of Hugh Woodin on ''singular cardinals hypothesis'' and '' Radin forcing''. Some of his results are published later by others, but it seems that there are still many ...

**6**

votes

**2**answers

652 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 ...

**17**

votes

**2**answers

1k views

### A set that can be covered by arbitrarily small intervals

Let $X$ be a subset of the real line and $S=\{s_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-small if there is a collection $\{I_i\}$ of intervals such that the length of ...

**15**

votes

**1**answer

902 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 ...

**13**

votes

**1**answer

690 views

### Sizes of bases of vector spaces without the axiom of choice

Assuming the axiom of choice does not hold we have that there is a vector space without a basis. The situation can be, in some sense, worse. It is consistent that there are vector spaces that have two ...

**10**

votes

**3**answers

2k views

### Kuratowski's definition of ordered pairs

Among several possible definitions of ordered pairs - see below - I find Kuratowski's the least compelling: its membership graph (2) has one node more than necessary (compared to (1)), it is not as ...

**9**

votes

**3**answers

787 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 ...

**7**

votes

**2**answers

2k views

### Definition of Function

What is authoritative canonical formal definition of function?
For example,
According to Wolfram MathWorld,
$$isafun_1(f)\;\leftrightarrow\;
\forall a\in f\;(\exists x\exists y \;\langle x,y\rangle ...

**7**

votes

**1**answer

184 views

### Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...

**7**

votes

**2**answers

416 views

### What can the degrees of constructibility be?

If $r, s\in\mathbb{R}$, we say $r$ is constructible relative to $s$ - and write $r\le_cs$ - if $r\in L[s]$. Modding out by the induced equivalence relation $\equiv_c$, we get a partial order, the ...

**4**

votes

**3**answers

223 views

### Approximation of infinite set in generic extension

Suppose $M$ is a c.t.m and suppose $P$ is $Fn(I,2)$ where $I$ is infinite. Now suppose $G$ is $P$-generic, and $A \in M[G]$ is infinite set.
Is it guaranteed that the exist $B \in M$ such that $B ...

**4**

votes

**2**answers

413 views

### Mutually generics

Given posets $P,Q\in M$, I would like to know under what circumstances there are mutually generic filters $G\subseteq P$ and $H\subseteq Q$ (generic over $M$). Also, what are the characterizations of ...

**10**

votes

**2**answers

545 views

### What are Moschovakis cardinals?

The question is exactly that of the title: what are Moschovakis cardinals?
Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency ...

**6**

votes

**0**answers

199 views

### Does Sageev's result need an inaccessible?

In 1981, building on work by Ellentuck in 1974, Sageev showed ("A model of ZF + there exists an inaccessible, in which the Dedekind cardinals constitute a natural non-standard model of arithmetic," ...

**5**

votes

**3**answers

713 views

### Why isn't there more interest in “large powerset axioms”?

By a large powerset axiom, let us mean informally an axiom that says that for some cardinal numbers $\kappa$, we have that $2^\kappa$ is somehow "large" or "difficult to access from below." For ...

**5**

votes

**0**answers

397 views

### The status of 'the consistency of NF relative to ZF'

One of the responses to my Mathoverflow question No. 122658 hinted that a proof (or the outline of a
proof) of the consistency of NF relative to ZF was on the horizon and was to be presented at a ...

**45**

votes

**15**answers

11k 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 ...