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.
1,112 questions
5
votes
0
answers
192
views
"Very $L$-like" models, part 1: large cardinals
(The original version of this question was much narrower and less natural; but see the edit history if interested.)
Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ ...
- 20.5k
5
votes
1
answer
2k
views
There are no abstract categories
$\newcommand\Set{\mathbf{Set}}\newcommand\Ob{\mathbf{Ob}}\newcommand\Hom{\mathbf{Hom}}$Work in a foundation that admits a countable hierarchy of notions of ‘set’, and say that a category is $n$-small ...
- 10.1k
5
votes
0
answers
265
views
How strong is this "modal definability principle"?
Throughout, we work in the class theory $\mathsf{MK}$ (although I'm open to tweaking this), "logic" means "set-sized logic whose semantics is definable over $V$," and "$\...
- 20.5k
5
votes
1
answer
183
views
Fragility of large cardinals with respect to transitive end extensions
To motivate things, let me start with a special case of the question I'm interested in. Let $\mathsf{In}(x)\equiv$ "$x$ is an inaccessible cardinal."
Question 1: Is it consistent with the ...
- 20.5k
5
votes
2
answers
1k
views
Large cardinals without the ambient set theory?
In an attempt to understand a bit better large cardinals, I have been thinking along the following lines, which could be summarized under the slogan
Talk about cardinals without the
(ambient) ...
- 7,853
5
votes
1
answer
158
views
(Weakly) minimal subcovers of linear covers
Motivation. The starting point of this question is the trivial observation that if we cover $\mathbb{N}$ with $$\big\{\{0,\ldots n\}: n\in \mathbb{N}\big\},$$ then this cover doesn't have a minimal ...
- 48.9k
5
votes
1
answer
1k
views
How many well-orders of reals are there?
It's commonly known that the cardinality of the set of all well-orders on $\aleph_0$ is the continuum (correct me if I'm wrong plz). What about that of all well-orders on $\mathbb{R}$? Is there a ...
- 77
5
votes
1
answer
422
views
What is the relationship between non-existence of those kinds of singular sets and AC?
Let's say a set $A$ is super-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $ |y| \not > |x|$ .
A set $A$ is ...
- 11.3k
5
votes
1
answer
149
views
Does there exist a section of $\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ that is "nearly Boolean"?
The following might be a somewhat esoteric question:
Does there exist an infinite cardinal $\kappa$ and a section $f$ of the quotient map $\pi:\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ (...
- 2,830
5
votes
0
answers
483
views
How to study formal logic without formally using the notion of a set?
I have recently begun curious in set theory, and when I researched this subject I saw that all axiomatizations of set theory, such as ZFC and NBG, are expressed in the language of first order logic. ...
- 331
5
votes
1
answer
286
views
Is there an oracle that can compute something iff it is computable in every countable model that is equivalent to $(V, \in)$?
Let us work in Kelly-morse set theory, so we can talk about $V$. For some model $M=(\mathbb N, \in_M)$ that is elementary equivalent $(V, \in)$, we can have an oracle that corresponds to $(\mathbb N, \...
- 6,353
5
votes
1
answer
471
views
Comparability implies well-orderability?
I am trying to prove a small proposition that got me completely stumped, and I cannot find a single counterexample.
(ZF) Suppose that $E$ is such that for every $A\subseteq\mathcal P(E)$ either $|...
- 39.8k
5
votes
0
answers
1k
views
Mathias forcing with Ramsey ultrafilters, and Cohen reals [closed]
Edit/update: One reason this question never received an answer is because it was founded on a faulty premise! The Blaszczyk-Shelah paper I mentioned below does not prove that Mathias forcing with a ...
- 2,701
5
votes
0
answers
449
views
How many Dedekind-finite sets can $\mathbb{R}$ be partitioned into?
Building off Asaf Karagila's answer to my previous question (Can $\mathbb{R}$ be partitioned into dedekind-finite sets?) on partitioning $\mathbb{R}$ into strictly Dedekind-finite sets:
(1) What ...
- 20.5k
5
votes
2
answers
528
views
Cardinality of a set of countable connected Hausdorff spaces
It is a non-trivial result that there is a countable connected Hausdorff space.
Let ${\cal T}$ be a set of connected Hausdorff topologies on $\omega$ such that whenever $\tau_1\neq\tau_2\in {\cal T}$ ...
- 48.9k
5
votes
1
answer
432
views
Is there a class choice principle over MK that is equivalent to class well ordering over MK?
$\sf MKCWO$ is the theory obtained by adding a new primitive binary relation $\prec$ to the signature of $\sf MK$ and axiomatize that $\prec$ is a well order on classes, that is:
$\textbf{Transitive:}...
- 11.3k
5
votes
1
answer
327
views
$\Sigma_n$ version of HOD
Fix a natural number, $n \geq 1$. Consider the class, M, of all sets hereditarily ordinal-definable using some $\Sigma_n$ formula. Since there is a universal $\Sigma_n$ formula, M is definable. Is M ...
- 1,174
5
votes
1
answer
421
views
Large cardinal axioms and the perfect set property
It is known that if there is a measurable cardinal then every $\Pi_1^1$ set has the perfect set property (i.e it is either countable or contains a copy of $2^{\omega}$). Also if we have $\Pi_1^1$-...
- 3,804
5
votes
2
answers
407
views
What's the consistency strength of this form of reflection?
Working in mono-sorted first order logic with equality $``="$ and membership $``\in"$:
Define: $set(x) \equiv_{df} \exists y \, (x \in y)$
Axiomatize:
Extensionality: $( a \subseteq b \land b \...
- 11.3k
5
votes
1
answer
493
views
Subcountability
In these slides of a talk Giovanni Curi shows that the generalized uniformity principle follows from Troesltra’s uniformity principle and from the subcountability of all sets, which are both claimed ...
- 411
5
votes
0
answers
472
views
The surreal numbers under a change of universe
Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\...
- 10.1k
5
votes
1
answer
489
views
Simplest non-constructible set of integers compatible with the nonexistence of $0^\sharp$?
What is the simplest non-constructible set of integers (say, in the analytical hierarchy) that is compatible with the nonexistence of $0^\sharp$? In particular, can there still be a non-constructible $...
- 4,985
5
votes
1
answer
627
views
Can there be no "surprisingly averageable" second-order sentences?
Say that a second-order sentence $\varphi$ is averageable iff there exists some infinite cardinal $\kappa$ and some nonprincipal ultrafilter $\mathcal{U}$ on $\kappa$ such that for every $\kappa$-...
- 20.5k
5
votes
3
answers
884
views
"name" for the ground model
I am a beginner of forcing, often I read from some articles something like "$p \Vdash \dot{G}$ is $P$-generic over $\check{M}$" (where $M$ is a countable transitive model, for instance).
Q1. I ...
- 53
5
votes
1
answer
694
views
What are examples of non-equivalent virtualizations of a large cardinal?
This is a follow up to my previous question concerning virtual large cardinals, that are generally weaker axioms of infinity obtained from ordinary large cardinals through the so-called virtualization ...
5
votes
1
answer
320
views
Bounds for a small cardinal
$\newcommand{\w}{\omega}\newcommand{\F}{\mathcal F}\newcommand{\I}{\mathcal I}\newcommand{\J}{\mathcal J}\newcommand{\M}{\mathcal M}\newcommand{\N}{\mathcal N}\newcommand{\x}{\mathfrak x}\newcommand{\...
- 5,409
4
votes
2
answers
883
views
Group & modules of arbitrary cardinality [closed]
How do I see that there is a group of an arbitrary cardinality? Is this also true for abelian groups? Also, given a commutative ring $R\neq 0$ how do I see that there is an $R$-module of arbitrary ...
- 2,857
4
votes
1
answer
302
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 ...
- 18.6k
4
votes
1
answer
309
views
Unbounded set in $V[G]$ has an unbounded subset in $V$?
This is a repost of the same question on math.SE, which received several comments but no answers/comments on the first question.
Suppose $\kappa$ is a cardinal preserved in the generic extension $V[G]...
- 1,432
4
votes
1
answer
318
views
Bad subforcings of nice forcing notions
Let $\mathbb{P}$ and $\mathbb{Q}$ be two forcing notions. Recall that we say $\mathbb{Q}$ is a subforcing of $\mathbb{P}$ if there exists a regular embedding $\mathbb{Q} \to \text{r.o.}(\mathbb{P}).$
...
- 32.1k
4
votes
1
answer
425
views
Perfect set property implies $\omega_1$ is a limit cardinal in $L$
Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega_1^V$ is actually a limit cardinal.
The original proof is in German, and I've been ...
- 39.8k
4
votes
0
answers
233
views
How distributive are the bad Laver tables?
Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,...,n\}$ and $*$ is the unique operation on $S_{n}$ where
$n*x=x$
$x*1=x+1\,\text{mod}\, n$ and
if $...
4
votes
0
answers
241
views
Is the lowenheim-skolem number of nth order logic larger than the corresponding number for 2nd order logic
According to this paper, by Vaananen, the $LS$ number for $2^{nd}$ order logic is given by "the supremum of $Π_{2}$-definable ordinals", where "The Lowenheim-Skolem number $LS(L)$ of $L$ is the ...
- 95
4
votes
1
answer
143
views
Injective choice function for infinite complete linear hypergraphs
A hypergraph $H=(V,E)$ is said to be complete and linear if
whenever $e_1\neq e_2\in E$ then $|e_1\cap e_2|=1$, and
for $v,w\in V$ there is $e\in E$ such that $\{v,w\}\subseteq e$.
Assuming that $V$ ...
- 48.9k
4
votes
0
answers
172
views
Ultracoproducts of C(X)-algebras
Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...
- 1,786
4
votes
1
answer
718
views
Is every element of $\omega_1$ the rank of some Borel set?
It is well known that we can obtain the $\sigma$-algebra of Borel subsets of $2^{\omega}$ in the following way: Let $B_0$ be the collection of all open subsets of $2^{\omega}$. For $\alpha=\beta+1$, ...
- 1,799
4
votes
1
answer
151
views
Comparing bornologies for domination/escaping
Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions from $\mathbb{N}$ to $\mathbb{N}$:
$\mathbb{D}=\{A: \exists f\in\mathcal{N}\forall g\in A\exists m\...
- 20.5k
4
votes
3
answers
828
views
Impact of the axiom of replacement on finite sets
The axiom of replacement is usually used to prove the existence of large sets, to provide a reflection principle, for transfinite recursion… However, I am wondering how it affects finite sets. Let me ...
- 2,442
4
votes
1
answer
260
views
Is ${\cal P}(\omega)/\text{(fin)}$ a fractal poset?
If $(P,\leq)$ is a partially ordered set and $a,b\in P$ we set $[a,b]:=\{x\in P: a\leq x\leq b\}$. We say that $P$ is fractal if whenever $a,b\in P$ and $[a,b]$ contains more than one element, then $[...
- 48.9k
4
votes
1
answer
746
views
Can all uncountable (but small) families of sets with positive measure have an uncountable subfamily with an intersection of positive measure?
My general question was is it consistent that any uncountable family of less than $\mathrm{non}(\mathcal{N})$ sets, each with positive measure, has an uncountable subfamily $\mathcal{F}$ s.t. $\bigcap ...
- 135
4
votes
1
answer
236
views
How can I collapse all cardinals of ground model except one of them?
Let $\kappa$ be an uncountable cardinal of a c.t.m $M$ of $ZFC$.
Is there a generic extension of $M$ like $M[G]$ such that all uncountable cardinals of ground model collapse except $\kappa$?
user44891
4
votes
0
answers
139
views
Simple $(\alpha+1)$-recursive well-orders with order type $|\alpha\text{-recursive}|$
In the following, $L_\alpha$ is the $\alpha$-th level of the constructible hierarchy, $\alpha$-recursive means definable in $L_\alpha$ by a $\Delta_1$ formula. $|\alpha\text{-recursive}|$ is the ...
- 148
4
votes
1
answer
221
views
Comparing bornologies for cardinal characteristics via Borel maps
This question is "take 2" of this older one, following a suggestion of Francois Dorais. Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions ...
- 20.5k
4
votes
2
answers
326
views
Infinite graph with no minimal vertex cover
If $G=(V,E)$ is a simple, undirected graph, then $C\subseteq V$ is said to be a vertex cover if $C\cap e\neq \varnothing$ for all $e\in E$.
Is there an infinite graph $G=(V,E)$ such that for any ...
- 48.9k
4
votes
1
answer
943
views
A question about "local" versus "global" large cardinal axioms
The terms "local" and "global" when applied to large cardinal axioms seem to have a well understood intuitive meaning, although a formalized definition of them in (a meta-language for)ZFC might be ...
- 6,215
4
votes
1
answer
888
views
About the relationship between the generalized continuum hypothesis and the axiom of choice
I was trying to get a short, intuitive proof of Sierpinski’s theorem (gch implies axiom of choice) and I could but only by using the following gch2 for the generalized continuum hypothesis gch.
gch: ...
- 481
4
votes
0
answers
297
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 $V[G]\models\phi(x_{1},...,...
4
votes
1
answer
556
views
the choice of representing formulas and Gödel's second incompleteness theorem
In Rautenberg's book (A Concise Introduction to Mathematical Logic, Universitext, Springer 2006), Gödel's second incompleteness theorem is stated:
Theorem 3.2 (Second incompleteness theorem). PA ...
- 143
4
votes
2
answers
448
views
Number of torsion-free abelian groups
Let $\mathfrak{c}$ be the cardinality of the continuum. How much Choice, if any, is needed to prove that there are $2^{\mathfrak{c}}$ distinct (mutually nonisomorphic) torsion-free abelian groups of ...
- 2,049
4
votes
2
answers
503
views
"Potentially club" filters on $\omega_2$
Short version: what can we say about subsets of $\omega_2$ which - in a generic extension where $\omega_2$ is the new $\omega_1$ - contain a club?
We could of course generalize beyond $\omega_2$, but ...
- 20.5k