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### Is the pseudomenon a statement? [migrated]

I'm asking this because I'm teaching a class on paradoxes for kids, and I realized I have no idea what the answer to this question is. It is a research question in the pedagogical sense, I suppose.
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317 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 ...

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456 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 ...

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### 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 ...

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401 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, ...

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403 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 ...

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380 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 $ ...

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### 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 ...

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760 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 ...

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### 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 $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...

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758 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 ...

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### “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 ...

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766 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 ...

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### 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 ...

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502 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 ...

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### 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 ...

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572 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 ...

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489 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 ...

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### 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.

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### 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 ...

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### 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 ...