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5
votes
1answer
242 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 ...
6
votes
1answer
340 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 ...
18
votes
1answer
477 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
1answer
230 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
3answers
405 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
1answer
426 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
2answers
391 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
5answers
807 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
3answers
772 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
2answers
466 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
3answers
765 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
3answers
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
4answers
784 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
3answers
653 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
2answers
518 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
5answers
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
2answers
578 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
2answers
502 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
2answers
415 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
3answers
327 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 ...
118
votes
26answers
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 ...