**7**

votes

**2**answers

487 views

### Independence of the countable axiom of choice

How does one proove that the Countable axiom of choice is not provable in ZF?Is there any brief proof?Does the Independence of the countable axiom of choice implies the independence of the axiom of ...

**5**

votes

**1**answer

291 views

### Relationship between fragments of the axiom of choice and the dependent choice principles

The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there ...

**7**

votes

**1**answer

367 views

### On Consistency of an Existence

Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement:
For all $ \mathfrak{B} \subseteq \mathbf{P}(\omega)$ with ...

**17**

votes

**1**answer

665 views

### Kaplansky's conjecture and Martin's axiom

Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...

**6**

votes

**1**answer

239 views

### Intuition behind Pincus' “injectively bounded statements”

In
David Pincus, Zermelo-Fraenkel Consistency Results by Fraenkel-Mostowski Methods,
The Journal of Symbolic Logic, Vol. 37, No. 4 (Dec., 1972), pp. 721-743
Pincus introduces the notion of ...

**3**

votes

**3**answers

413 views

### Undecidability and holomorphic functions (Reference request)

The goal of this question is to recall a certain mathematical fact -not in my field- that I was once briefly told and that I have fogotten, and also to collect similar results.
The fact, I think, ...

**14**

votes

**2**answers

534 views

### Nice Algebraic Statements Independent from ZF + V=L (constructibility)

Background and Motivation
I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering ...

**13**

votes

**2**answers

422 views

### Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.

This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes.
Let $ ...

**6**

votes

**5**answers

842 views

### Formal proof of Con(ZFC) => Con(ZFC + not CH) in ZFC

Is it possible to prove $Con(ZFC) \rightarrow Con(ZFC + \neg CH)$ purely within ZFC? To prove this (using forcing) one seems to need a countable transitive model of ZFC. The texts I am reading avoid ...

**8**

votes

**3**answers

802 views

### Is the Axiom of Union independent of the rest of ZF?

Short version: Is the axiom of union independent of the rest of axioms of ZF?
NO) Tourlakis (2003) says in p. 177 that the axiom of union can be derived from the rest of ZF if an appropriate version ...

**6**

votes

**2**answers

475 views

### Natural statements independent from true $\Pi^0_2$ sentences

I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0_2$ consequences of true arithmetic $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...

**4**

votes

**3**answers

776 views

### Is there an “undecided” assertion of which a proof that it's not undecidable is known?

Just a curiosity:
Is there an assertion of which a proof (formalizable, say, in ZFC) is not known but a proof that it's not undecidable (in ZFC) is known?
Edit: after the comments, I think the ...

**26**

votes

**3**answers

2k views

### “Simpler” statements equivalent to Con(PA) or Con(ZFC)?

Given any reasonable formal system F (e.g., Peano Arithmetic or ZFC), we all know that one can construct a Turing machine that runs forever iff F is consistent, by enumerating the theorems of F and ...

**10**

votes

**4**answers

824 views

### Is every p-point ultrafilter Ramsey?

A non-principal ultrafilter $\mathcal{U}$ on $\omega$ is a p-point (or weakly selective) iff for every partition $\omega = \bigsqcup _{n < \omega} Z_n$ into null sets, i.e each $Z_n \not \in ...

**6**

votes

**3**answers

686 views

### Reference Request: Independence of the ultrafilter lemma from ZF

I'm looking for references for the following facts concerning the ultrafilter lemma (~ "there exist non-principal ultrafilters"):
The ultrafilter lemma is independent of ZF.
ZF + the ultrafilter ...

**8**

votes

**2**answers

559 views

### Statements forced by one condition of a poset, but not the whole thing

In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be ...

**11**

votes

**5**answers

2k views

### Minimal subset of axioms for ZFC

Hello all, one may look for "minimal system of axioms" for ZFC (or any other
theory) in the following (unusual) sense : say that a subset S of ZFC is
"sufficient" if there is an explicit procedure ...

**6**

votes

**2**answers

601 views

### Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?

It's not hard to show that for any cardinal $\kappa$, there is no dense linear order without endpoints (DLO) of size greater than $2^{\kappa}$ that has a dense subset of size $\kappa$. But one can ...

**2**

votes

**2**answers

512 views

### A problem about posets similar to Suslin's problem

Suslin's problem is:
Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$?
The answer is that it's independent of ZFC. The related ...

**2**

votes

**2**answers

419 views

### Is the existence of a well-ordering on R independent of ZF?

I am reasonably certain this is the case, but can't find a reference that actually states this, although the Wikipedia article states something close.

**3**

votes

**3**answers

328 views

### Is there a formula phi s.t. phi and not-phi have a stronger consistency?

Let Σ be an axiom system. Can there be a formula φ, s.t.
Con(Σ) does not imply Con(Σ + φ) AND
Con(Σ) does not imply Con(Σ + not φ)
If yes, can you give ...

**122**

votes

**26**answers

15k 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 ...