**11**

votes

**1**answer

275 views

### A classic cardinal characteristic of the continuum in disguise?

We believe the answer to the following question, that is relevant to a joint research project with Piotr Szewczak, should be known. We would appreciate any help or pointer.
Needed definitions may be ...

**8**

votes

**3**answers

358 views

### Freiling's Axiom of Symmetry Concretized

Freiling's Axiom of Symmetry says that for any function $f:[0,1]\to \mathcal{P}([0,1])$ such that for every $x\in [0,1]$ we have $|f(x)|=\aleph_0$, then there exist $y,z\in [0,1]$ such that $z\notin ...

**13**

votes

**4**answers

2k views

### Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$?

In a conversation where it came up that the Pythagoreans probably found an enumeration of the rational numbers I erroneously remarked that Georg Cantor found a natural bijection from $\mathbb{N}$ to ...

**3**

votes

**0**answers

177 views

### What algebraic identities does the iteration of forcing operation satisfy?

Let $G$ be the set of all formulas $\phi(x)$ in the language of such that $ZFC\vdash\exists x\phi(x)$ exists, $ZFC\vdash\phi(x)\rightarrow``x\,\textrm{is a complete Boolean algebra}"$, ...

**4**

votes

**1**answer

211 views

### $\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ and its implication

A. S. Daghighi, M. Golshani, J. D. Hamkins, and E. Jeřábek proved in "The foundation axiom and elementary self-embeddings of the universe" that, working in ZFGC$^{\text{−f}}$+BAFA, there are ...

**8**

votes

**2**answers

318 views

### For a partition of $\mathbb{R}$ into countably infinite sets, must there be an almost-disjoint family of $2^{\frak c}$ many selectors?

My question arises from a construction I gave in my recent
answer to a question of Alexander Pruss concerning large families of independent non-measurable sets of reals. In that argument, using
the ...

**4**

votes

**0**answers

228 views

### Is there some absoluteness between the Boolean valued universe $V^{B}$ and $V$?

It is well known that if $\phi$ is a $\Delta_{1}$-formula and $x_{1},..,x_{n}$ in $V$ and $V[G]$ is a forcing extension, then $V\models\phi(x_{1},...,x_{n})$ if and only if ...

**9**

votes

**3**answers

308 views

### Images of $\{0,1\}^\kappa$

Is there a compact topological space $(X,\tau)$ such that for no cardinal $\kappa$ there is a surjective continuous map $e:\{0,1\}^\kappa \to X$?
(We assume that $\{0,1\}$ is endowed with the ...

**0**

votes

**2**answers

124 views

### Surjectivity from union of a set system to the set system

Let $\mathcal{A}$ be a non-empty systems of non-empty sets such that there is an injective map $f:\bigcup \mathcal{A}\to \mathcal{A}$ such that $a\in f(a)$ for all $a\in\bigcup\mathcal{A}$. Assuming ...

**2**

votes

**0**answers

123 views

### Surjective marriages

Let $M, W\neq \emptyset$ be sets and $K\subseteq M\times W$. We say that $(M, W, K)$ has a marriage if there is an injective function $f:M\to W$ such that $f\subseteq K$.
If $(M,W, K)$ has a ...

**8**

votes

**1**answer

342 views

### If $\kappa$ is weakly inaccessible and $A\subset\kappa$, can $L[A]$ violate $\kappa^{\lt\kappa}=\kappa$?

In some current work, my co-authors and I had wanted in a certain
argument to appeal to $\kappa^{\lt\kappa}=\kappa$ in $L[A]$, in a
situation where $A\subset\kappa$ and $\kappa$ was weakly
...

**6**

votes

**1**answer

185 views

### Surjective (strong) reducibility of Borel equivalence relations

Suppose $E$ and $F$ are Borel equivalence relations on Polish spaces $X$, $Y$, resp. Say that $E$ is surjectively Borel reducible to $F$ iff there is a Borel surjection $f:X \to Y$ such that $xEy$ iff ...

**4**

votes

**1**answer

331 views

### Inaccessible cardinal and $\Sigma_1$ reflection

A theorem of A. Levy says that, if $\kappa$ is an inaccessible cardinal, then $V_\kappa\prec_{\Sigma_1}V$ namely $V_\kappa$ is an elementary submodel when considering only $\Sigma_1$ formulas.
Where ...

**7**

votes

**2**answers

529 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 ...

**4**

votes

**1**answer

129 views

### $\mathfrak{p}=\mathfrak{b}=\mathfrak{a}=\aleph_1$ and $\mathfrak{d}=\mathfrak{c}=\kappa$

Asumme tha in $M$, $CH$ holds and $\kappa>\aleph_0$ and $\kappa^{\aleph_0}=\kappa$. Let $K$ be $Fn(\kappa,2)$-generic over $M$.
Question:
Then we can say in $M[K]$ that:
$(i)$ ...

**3**

votes

**1**answer

224 views

### Hamkins infinite time Turing machines: dovetailing ordinal time

It is claimed in the Hamkins and Lewis founding article "Infinite time Turing machines" (proof of the gap existence theorem 3.4) that for $\omega$ steps of a computation of a machine performing a ...

**6**

votes

**0**answers

122 views

### $\kappa$-impediments (according to Shelah, Nash-Williams, Aharoni)

Let $\Gamma = (M, W, K)$ be a bipartite graph, that is $M, W$ are sets and $K\subseteq M\times W$. If there is an injective function $f:M\to W$ such that $f\subseteq K$ we say $f$ is an espousal and ...

**2**

votes

**0**answers

288 views

### A query on how to climb inaccessibles in £

I am investigating to what extent extensions of the librationist property or set theory £ may support relative inaccessible sets; see Librationist Closures of the Paradoxes and Elements of ...

**2**

votes

**1**answer

112 views

### “Namba forcing adds reals” independent of $ZFC + \neg CH$?

I know that, in the presence of $CH$, Namba forcing does not add reals. But when $CH$ fails, is it consistent that it still does not add reals?

**14**

votes

**2**answers

578 views

### Scott-Solovay unpublished paper on ``Boolean valued models of set theory''

I have read some papers from 1970$^{th}$, and in some of them, the paper of Scott and Solovay on ``Boolean valued models of set theory'' is given as a main reference, with many references to the ...

**6**

votes

**3**answers

255 views

### Borel cross section

It is known from metric space topology that a closed equivalence relation on a Polish space has either countably many or $\mathfrak{c}$ many equivalence classes.
A short elementary proof is given in ...

**3**

votes

**3**answers

856 views

### Reverse Skolem's paradox

By using the Löwenheim–Skolem theorem & Mostowski collapse, in every model $V$ of $ZF+Con(ZF)$ there is a countable transitive set $M$ such that $(M,\in_M) \models ZF$. Is the following "converse" ...

**17**

votes

**2**answers

1k views

### Philosophical arguments in defense (or against) large cardinals

The question is essentially what is asked in the title. I split it into two parts
(A) (Arguments supporting the existence of large cardinals) What are the main philosophical arguments in defense ...

**5**

votes

**1**answer

201 views

### Are Sharps in Countable Models Really Sharps?

Suppose $M$ is a countable transitive model of $\mathsf{ZFC}$ (maybe more). Suppose $x, y \in {}^\omega\omega \cap M$ and $M \models y = x^\sharp$. Is it true (possible with additional assumptions), ...

**8**

votes

**2**answers

242 views

### Is there a version of Miller forcing “guided by” an ultrafilter?

It’s well known that Mathias forcing factors as a two-step iteration $P(\omega)/\mathrm{fin}\ast \mathbb{M}_{\dot U}$, where $\mathbb{M}_{\dot U}$ is Mathias forcing guided by the generic ultrafilter ...

**11**

votes

**1**answer

485 views

### Continuous functions and 2-bushy trees

The following problem was asked by Joe Miller in the fall of 2010 at a bar in Madison.
A subtree $T \subseteq 4^{< \omega}$ is $2$-bushy if for some node $\sigma \in T$, every node above $\sigma$ ...

**16**

votes

**6**answers

865 views

### Application of Fraïssé construction in set theory

As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property.
Now I would like to know ...

**6**

votes

**0**answers

233 views

### Quasi-disjoint subsets of an infinite set and $\neg \mathsf{AC}$

Is it consistent with $\mathsf{ZF}$ (without $\mathsf{AC}$) that there is an infinite set $X$ and a subset $S\subseteq\mathcal P(X)$ of the same cardinality as $\mathcal P(X)$ with the property that ...

**13**

votes

**2**answers

1k views

### Do the real numbers “know” that they are countable in a larger model?

(This was first posted to math.stackexchange but had no answers there after several days):
Let ${\mathbb R}$ be the set of real numbers in whatever is your favorite model of $ZFC$. Then (by Levy ...

**8**

votes

**0**answers

167 views

### History of preservation theorems in forcing theory

For my honours thesis, I am studying a general preservation theorem using a framework provided by Shelah. I am mainly concerned about revised countable support iteration of $\dot{S}$-semiproper ...

**1**

vote

**1**answer

125 views

### Questions about a possible way of representing construcive ordinal numbers

Let $K$ be the set of all total recursive functions of non-negative integers having only non-negative integers as values. Let $L$ be any well-ordered subset of $K$ in which the ordering $<$ is ...

**6**

votes

**1**answer

452 views

### Adding sets not containing arithmetic progressions of length three by forcing

Consider the following forcing notion: conditions in $\mathbb{P}$ are pairs $(s, N),$ where:
1) $s\in 2^{<\omega}$,
2) $N\in \mathbb{N}$,
3) (by identifying $s$ with a subset of $lh(s)$) $s$ ...

**6**

votes

**0**answers

197 views

### Tree property and singular strong limit cardinals

I heard that the following theorem is proved recently by Foreman-Magidor, which answers a famous old open question:
Theorem. It is consistent, relative to the existence of large cardinals, that ...

**9**

votes

**2**answers

471 views

### Non-Forcing and Independence

I asked this question about two weeks ago on MSE and haven't gotten an answer, so I thought I would post the question here.
Do there exists sentences which are independent of ZFC, cannot be shown to ...

**8**

votes

**1**answer

241 views

### Translates of meager sets

Does there exist a meager set of reals M such that every meager set can be covered by countably many translates of M? This is the category analogue of the following.

**-2**

votes

**1**answer

185 views

### Forcing and $\mathbb{P}$-name [closed]

If $\mathbb{P}$ be a poset, $\dot{Q}$ a $\mathbb{P}$-name and $\mu$ an infinite cardinal such that $\Vdash 0<|\dot{Q}|\leq\mu$.
$(a)$ Exist names $\langle \dot{q}_\alpha\rangle_{\alpha<\mu}$ ...

**1**

vote

**1**answer

244 views

### Confusion with proof about a fact $\mathbb{P}$-name [closed]

Let $\mathbb{P}$ be poset.
Let $B$ be a set. We say that a $\mathbb{P}$-name $\dot{b}$ is a nice name for member of $B$ if there is a maximal antichain $A\subseteq\mathbb{P}$ and a function ...

**2**

votes

**2**answers

247 views

### Zorn's lemma via Zermelo theorem [closed]

Is there a way to deduce Zorn's lemma from Zermelo theorem (that any set may be well ordered), which is essentially shorter then deduction of Zorn's lemma from the usual form of Axiom of Choice?

**31**

votes

**2**answers

796 views

### Translates of null sets

Does there exist a null set of reals $N$ such that every null set is covered by countably many translations of $N$?

**2**

votes

**1**answer

140 views

### Ore's theorem for countable graphs

Ore's theorem states that in a finite graph $G$ with $|V(G)|=n$, there is a Hamiltonian path, provided that the sums of the degrees of 2 distinct, non-adjacent vertices is $\geq n$.
For countable ...

**6**

votes

**1**answer

293 views

### Axiom of choice and the equality between second-order constructible universe and HOD

I try to prove $L_{SO}=\mathrm{HOD}$, where $L_{SO}$ is second-order constructible universe which has similar definition with $L$ but it uses second-order definability rather than the first-order ...

**4**

votes

**1**answer

143 views

### presaturated ideals

In this paper, Gitik and Shelah make the following claim (part of Proposition 1.5):
Claim (Gitik-Shelah): Suppose $\kappa < \lambda$ are regular, $2^\lambda = \lambda^+$, and $D$ is a normal ...

**5**

votes

**2**answers

542 views

### A specific Model of ZFC

In his paper "Some Second Order Set Theory", Joel Hamkins asked whether there is a model of set theory $V$ that is elementary equivalent to $V[G]$, Whenever $G$ is $V$-generic for the collapse of a ...

**4**

votes

**2**answers

246 views

### Proving results about complete Boolean algebras in ZFC using Boolean valued models

I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing.
I am mainly ...

**6**

votes

**1**answer

202 views

### $\text{Cont}(X,X)$ and $\neg\mathsf{GCH}$

For a topological space $(X,\tau)$ let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$. Is it consistent that there a space $(X,\tau)$ such that $$|X| < |\text{Cont}(X,X)| < ...

**3**

votes

**1**answer

92 views

### The GCH in a reverse Easton support iteration

I am trying to understand the proof that the GCH can first fail at a weakly compact cardinal. We assume the GCH and that there exists a weakly compact cardinal $\kappa$, and we construct a reverse ...

**3**

votes

**4**answers

367 views

### Topological spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$

Let $(X,\tau)$ be a topological space. Let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$.
What can be said about spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$? For ...

**4**

votes

**1**answer

280 views

### stationary tower forcing

It is known that if $\delta$ is a Woodin cardinal and $\kappa < \delta$, then the stationary tower forcing $\mathbb Q^\kappa_{<\delta}$ preserves cardinals up to $\kappa$ and forces $\delta = ...

**6**

votes

**1**answer

230 views

### Is there a simple combinatorial characterization for when a direct limit of ultrapowers of $V$ is well-founded?

I want to know if there are fairly simple combinatorial necessary conditions for when a direct limit of ultrapowers of $V$ is well-founded similar to $\sigma$-completeness. By combinatorial, I mean ...

**6**

votes

**0**answers

168 views

### Singular Jonsson cardinals

Is the consistency of the following well-known:
$(*)$: There exists a singular cardinal $\kappa$ such that :
(1) $\kappa$ is a Jonsson cardinal,
(2) $\kappa$ is not a fixed point of the ...