Questions tagged [set-theory]
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
5,256
questions
32
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2
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1k
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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
281
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
564
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
292
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 ...
6
votes
2
answers
860
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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
526
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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 ...
3
votes
1
answer
336
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)| < ...
4
votes
1
answer
278
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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
521
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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 ...
6
votes
1
answer
980
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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 = \...
7
votes
1
answer
359
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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 ...
12
votes
0
answers
361
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Singular Jonsson cardinals
Is the following consistent?
$(*)$: There exists a singular cardinal $\kappa$ such that :
(1) $\kappa$ is a Jonsson cardinal,
(2) $\kappa$ is not a fixed point of the $\aleph-$function, i.e., $\kappa ...
7
votes
2
answers
251
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 ...
14
votes
1
answer
762
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 ...
7
votes
1
answer
240
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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
273
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
292
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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
110
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
429
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 ...
20
votes
1
answer
2k
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 ...
13
votes
2
answers
622
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
251
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 P_\...
1
vote
1
answer
101
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
1k
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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 B)...
7
votes
1
answer
886
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
890
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 ...
4
votes
0
answers
144
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]$ ...
17
votes
3
answers
5k
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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 ...
2
votes
1
answer
340
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 $\text{Part}(...
19
votes
2
answers
2k
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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 ...
6
votes
3
answers
514
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 ...
10
votes
1
answer
824
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 ...
7
votes
0
answers
289
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
173
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 P,S\...
7
votes
0
answers
429
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
328
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
749
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
476
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 ...
15
votes
4
answers
1k
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 ...
6
votes
1
answer
805
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
897
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
571
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 ...
1
vote
1
answer
120
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 C$...
15
votes
2
answers
760
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
405
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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
285
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 ...
9
votes
1
answer
401
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 ...
4
votes
1
answer
240
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 ...
34
votes
5
answers
2k
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 ...
12
votes
1
answer
680
views
A new cardinality living in every forcing extension?
I'm broadly interested in notions of "generic presentability" - when a given object exists in every forcing extension of the universe by some fixed forcing, at least up to the appropriate ...