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4
votes
0answers
177 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 ...
11
votes
2answers
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
0answers
130 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
1answer
114 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
1answer
402 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 $lh(s)$) $s$ ...
5
votes
0answers
145 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
2answers
410 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
1answer
215 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.
-6
votes
0answers
78 views

Can you give me some example of each these definition [closed]

I'm learning a partially ordered set.Can you give me some example of each these definition: 1.Upper and Down Bound : ...
-4
votes
0answers
77 views

every(ultra)filter on set I is principle if and only if I is finite [closed]

1)the filter generated by{a,b} is not ultra filter? 2)the filters generated by singleton are precisely the principle ultrafilters. 3)every(ultra)filter on set I is principle if and only if I is ...
-2
votes
1answer
170 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
1answer
207 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
2answers
222 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?
30
votes
2answers
683 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
1answer
130 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
1answer
269 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
1answer
130 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
2answers
418 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 ...
1
vote
0answers
102 views

Closed subsets of Souslin orders [migrated]

A total order without end-points is Souslin if it is complete, non-separable and ccc. Such orders may or may not exist but when they do, there can be a vast zoo of them. Can we get consistent ...
4
votes
2answers
192 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
1answer
192 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
1answer
84 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
4answers
355 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
1answer
246 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
1answer
219 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
0answers
156 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
1answer
142 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
1answer
513 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
1answer
141 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
0answers
207 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
1answer
207 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
0answers
96 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
1answer
302 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
1answer
422 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
1answer
381 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
1answer
194 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
1answer
81 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
4answers
216 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
1answer
248 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
1answer
434 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
0answers
113 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
2answers
300 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
1answer
109 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
1answer
402 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
2answers
263 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
1answer
349 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
0answers
166 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 ...
8
votes
0answers
103 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
0answers
192 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
2answers
274 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 ...