**30**

votes

**2**answers

731 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 $N$?

**0**

votes

**1**answer

132 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

275 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

132 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

433 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

204 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 ...

**5**

votes

**1**answer

195 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

88 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 ...

**2**

votes

**4**answers

359 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

259 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

225 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

162 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 ...

**7**

votes

**1**answer

147 views

### $2$-uniformization versus $\omega$-uniformization of ladder systems

Let $S \subseteq \omega_1$ be a stationary set of limit ordinals and let $L = \langle L_\alpha \;|\; \alpha\in S\rangle$ be a ladder system. We say that $L$ has $\kappa$-uniformization if for every ...

**12**

votes

**1**answer

523 views

### Ordinary mathematics in Chang's model

This question is prompted by a paper by Andre Kornell that just appeared on the arXiv. A large portion of the paper is devoted to showing that a surprising amount of ordinary mathematics can be ...

**5**

votes

**1**answer

143 views

### Pseudo-Prikry sequences vs Prikry sequences

Definition: Let $V\subseteq W$ be two transitive models of $ZFC$. A pseudo-Prikry sequence, $s$, at a cardinal $\kappa$ for $(V, W)$ is an $\omega$-sequence, cofinal at $\kappa$ such that for every ...

**9**

votes

**0**answers

211 views

### Proving regularity properties from forcing axioms

It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc.
Is there a direct proof ...

**4**

votes

**1**answer

214 views

### Sharps and Every Set is Constructible from a Real

Is it consistent that there is a model of $\mathsf{ZFC}$ (or $\mathsf{ZF}$) with the following properties:
(1) For all $x \in {}^\omega 2$, $x^\sharp$ exists (or $\mathbf{\Sigma}_1^1$ determinacy)
...

**2**

votes

**0**answers

97 views

### May $\Sigma_3$-collection hold below $\Sigma_3$-admissible ordinals for Gödel's L?

Suppose you have a system X=$KP$ + infinity plus $\Sigma_{3}$-collection and $\Delta_{2}$-specification. May $L_{\delta}\vDash X$ for some $\delta$ smaller than all $\Sigma_{3}$ admissible ordinals?

**9**

votes

**1**answer

311 views

### singularize the least inaccessible?

Is it consistent that there is some partial order $\mathbb P$ and some inaccessible cardinal $\kappa$, which is the least inaccessible, such that $\mathbb P$ forces $\kappa$ to be singular while ...

**13**

votes

**1**answer

435 views

### Can ZFC prove it cannot derive an inconsistency in $n$ steps?

Let $Con(\mathtt{ZFC}, n)$ denote the statement "$\mathtt{ZFC}$ cannot prove the contradiction within $n$ steps (or better within $n$ symbols) within a given proof system (say a natural deduction to ...

**8**

votes

**1**answer

384 views

### Adding a real with infinite conditions

Consider the forcing $\Bbb P$ whose conditions are partial functions $p\colon\omega\to2$ with $\operatorname{dom}(p)$ a co-infinite subset of $\omega$, ordered by reverse inclusion.
Does $\Bbb P$ ...

**7**

votes

**1**answer

196 views

### Generic filters of inverse limits

Maybe this doubt is silly, but I do not understand the final step of the proof of Lemma 5.2 in Hamkins' paper Fragile measurability, Journal of Symbolic Logic 59 (1994) 262-282.
There, $\mathbb ...

**1**

vote

**1**answer

82 views

### On whether a formula of KP is $\Pi_3$

In the context of KP, is the formula $\forall w(w\in x \leftrightarrow\forall y\exists z F(w,y,z))$ $\Pi_3$ when $F(w,y,x)$ is $\Delta_0$?

**0**

votes

**4**answers

228 views

### Approximating an arbitrary $\sigma$-algebra by simpler $\sigma$-algebras

A $\sigma$-algebra $\mathcal F$ over $\Omega$ is generated by an countable partition if there exits a countable partition $\mathcal B = \{ B_i \}$ of $\Omega$ such that $\mathcal F = \sigma(\mathcal ...

**6**

votes

**1**answer

255 views

### Infinite graphs isomorphic to their line graph

The only finite connected graphs $G$ that are isomorphic to their line graph $L(G)$ are the cycle graphs $C_n$ (see this link for example).
There are connected countable graphs that are isomorphic to ...

**5**

votes

**2**answers

546 views

### Consequences of ZF+“all subsets of reals are Lebesgue measurable”

(I'm not sure if this is entirely suitable here so feel free to close it if it's not.) The statement "there is a Lebesgue measure on $\mathbb{R}$($2^\omega$)" means: there is a total $\sigma$-additive ...

**3**

votes

**0**answers

117 views

### a game with generic filters

The present question is a follow-up to this one. Assume GCH holds in $V$ and suppose $G \subseteq Add(\omega_1,\omega_2)$ is generic over $V$. For any $g \subseteq Add(\omega_1,\alpha)$ in $V[G]$ ...

**10**

votes

**2**answers

317 views

### Is there a compendium of the consistency strength between the most important formal theories?

Preliminar Notions:
A formal system is a tuple $(\Sigma,G,A,R)$ where $\Sigma$ is an alphabet (set of symbols), $G$ is a formal grammar on $\Sigma$ that generates a formal language $L$ (set of well ...

**3**

votes

**1**answer

113 views

### Infimum of partitions

Let $X\neq\emptyset$ be a set. A partition is a subset $P\subseteq {\cal P}(X)\setminus \{\emptyset\}$ such that $\bigcup P = X$ and any distinct members of $P$ are disjoint. We denote by ...

**11**

votes

**1**answer

466 views

### Analogues of Primitive Recursive Functions

Let $\mathbf{A}$ be an admissible set (possibly with urelements). I am wondering if there is some good notion of "primitive recursive arithmetic" relative to $\mathbf{A}$. More precisely, I would like ...

**4**

votes

**2**answers

268 views

### Limits of determinacy on reals

For $X\subseteq\mathbb{R}^\omega$, say that $X$ is determined if the associated game on $\mathbb{R}$ of length $\omega$ (players I and II alternate playing reals, player I wins iff the sequence built ...

**7**

votes

**1**answer

357 views

### Is the functor of points of a scheme cofinally small?

Background: In functorial algebraic geometry one would like to consider the category of all functors $\mathsf{CRing} \to \mathsf{Set}$ and define/characterize the category of schemes as a full ...

**6**

votes

**0**answers

167 views

### Core model for supercompact cardinals and iteration trees

I have a few somehow related questions:
Question 1. What do we expect for an inner model $\mathcal{K}$ to be a core model for a supercompact cardinal? What properties should it have, and what ...

**9**

votes

**0**answers

108 views

### Is it possible that all ultrafilters are determined by the meet-semilattice of sub-ultrapowers?

Suppose that $\mathcal{Z}$ is a filter on a set $X$. Let $\Pi(X)$ denote the lattice of all partitions of the set $X$. Then $(\Pi(X),\wedge)$ is a meet-semilattice where
$P\wedge Q=\{R\cap S|R\in ...

**6**

votes

**0**answers

199 views

### $\delta$-strong compactness and generalized strong tree properties

Are there non-trivial equivalent characterizations of $\delta$-strongly compact (and almost strongly compact) cardinals in terms of generalized tree properties?
Recall the definitions as per Joan ...

**5**

votes

**2**answers

277 views

### The 'class version' of almost disjoint sets: can it fail?

I have a question about 'class versions' of almost disjoint sets. To even state what I'm after, I need to go beyond what one can state in theories like NBG or MK. I'm wondering about the status ...

**3**

votes

**1**answer

247 views

### Cardinality of an ultraproduct

Given structures $A_i$ each of cardinality $<\kappa$ where $\kappa$ is a measurable cardinal, the cardinalities of the $A_i$ are not uniformly bounded by a cardinal $\lambda <\kappa$, and ...

**6**

votes

**1**answer

217 views

### Are two forms of the Dual Schroeder-Bernstein property equivalent?

We know the Shroeder-Bernstein (SB) theorem can be proved in ZF, while the Dual Schroeder-Bernstein (DSB) can be proved in ZF+AC but not in ZF. Define as ISB the property that whenever there are both ...

**13**

votes

**3**answers

524 views

### Products of Cohen forcings

Let $\lambda$ be an infinite cardinal. Does the full support product of $\lambda$ copies of $Add(\omega, 1)$ collapse $2^\lambda$ to $\aleph_0$?
For $\lambda = \omega$, it is known to be true (it is ...

**4**

votes

**1**answer

419 views

### Did Brouwer evade uncountability?

I have the distinct memory of having often heard and read that intuitionism was inter alia geared to avoid Cantor's uncountable sets, and it may be that this was Brouwer's plan. But are there accounts ...

**1**

vote

**3**answers

583 views

### Brouwer vs. Cantor

Brouwer criticises Cantor e.g. in Intuitionistiche Mengenlehre. Is there a link or reference to some streamlined modern account of Brouwer's ideas?

**2**

votes

**1**answer

338 views

### A question regarding the Hahn-Banach theorem

Wikipedia states that, in $ZF$, the Axiom of Choice ($AC$) implies the Hahn-Banach theorem, but that the Hahn-Banach theorem does not imply $AC$. It also states that in $ZF$, the Hahn-Banach theorem ...

**2**

votes

**1**answer

95 views

### Minimality condition in a certain class of hypergraphs

A hypergraph is a pair $H=(V,E)$ such that $V$ is a (possibly infinite) set and $E\subseteq \mathcal{P}(V)$. $C\subseteq E$ is said to be a cover if $\bigcup C = V$ and $C$is minimal if $C'\subseteq ...

**14**

votes

**2**answers

461 views

### Preservation of properness

Suppose $\mathbb P$ is a countably closed forcing, and $\mathbb Q$ is a c.c.c. forcing that adds reals. Is $\mathbb P$ still proper in $V^{\mathbb Q}$?

**3**

votes

**1**answer

264 views

### Some questions on (non)-measurable sets without AC

In his answer to a Math Stack Exchange question of Katlus, Asaf Karagila wrote the following:
"It is a theorem that from $ZF+DC+$"$\aleph_1$$\le$$|$$\mathbb R$$|$" we can prove that there is an ...

**11**

votes

**1**answer

217 views

### counterexample regarding quotient algebras in forcing

Suppose $A$ and $B$ are complete subalgebras of a complete boolean algebra $C$. Let $G \subseteq A$ be generic. In the extension $V[G]$, we can define the quotient algebras $B/G$ and $C/G$ in the ...

**8**

votes

**1**answer

248 views

### What is known about global well ordering of classes in Gödel-Bernays?

I would like to have something like a linear order on classes, such that every instantiated predicate of classes has a minimal instance in that order. For my purposes, it is fine to assume V=L for ...

**5**

votes

**1**answer

123 views

### Strongly minimal covers

Let $H=(V,E)$ be a hypergraph, that is $V$ is a set and $E\subseteq \mathcal{P}(V)$. We say that $C\subseteq E$ is a cover of $H$ if $\bigcup C = V$.
A cover $M\subseteq E$ is said to be strongly ...

**23**

votes

**3**answers

843 views

### Forcing as a replacement of induction and diagonal arguments

Let me give some examples motivating the question.
The use of forcing instead of induction: For this consider Cantor's theorem:
Theorem 1. Any two countable dense linear orders $I, J$ without end ...

**6**

votes

**0**answers

228 views

### A new cardinality living in every forcing extension?

This question is motivated by the papers http://arxiv.org/abs/1405.7456 and http://arxiv.org/abs/1410.1224.
Say that a set $X$ is "generically presentable" over $V\models ZF$ if there is some ...