# Tagged Questions

**5**

votes

**1**answer

226 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 ...

**5**

votes

**1**answer

321 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

**1**answer

462 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

**5**answers

797 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 ...

**6**

votes

**2**answers

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

**8**

votes

**2**answers

506 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 ...

**3**

votes

**2**answers

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 ...