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.
5,255
questions
4
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2
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141
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Are there outer models $V \subset W$ of $L$ such that $V$ is "far" from $L$ but $W$ is "not too far" from $V$?
In the following, whenever I say "$V_1$ is an outer model of $V_2$", I mean
$V_1, V_2$ are transitive models of $\mathsf{ZFC}$,
$V_2 \subset V_1$,and
$ORD^{V_1} = ORD^{V_2}$.
I am curious ...
2
votes
1
answer
221
views
A possible ${\sf (ZF)}$-theorem in the spirit of the $3$-set-lemma
The number $3$ plays an interesting role in the following statement:
$\newcommand{\S}{\sf(S_3)}\S$ Let $X$ be a non-empty set and let $f:X \to X$ be fixpoint-free (that is $f(x) \neq x$ for all $x\in ...
5
votes
1
answer
198
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On the number of complete Boolean algebras
In their 1972 paper On the number of complete Boolean algebras Monk and Solovay showed that if $\lambda$ is an infinite cardinal, then there are $2^{2^\lambda}$ many isomorphism types of
complete ...
4
votes
1
answer
193
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Weak Power Hypothesis and Dependent Choice
Consider in $\newcommand{\ZF}{{\sf (ZF)}}\ZF$ the following statement:
Weak Power Hypothesis (WPH): if $X,Y$ are sets and there is a bijection between $\newcommand{\P}{{\cal P}}\P(X)$ and $\P(Y)$, ...
4
votes
1
answer
169
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Generic absoluteness
In Theorem 14 of "On The Question Of Absolute Undecidability" Peter Koellner describes a generic absoluteness result which could be summed up as $\Sigma^2_1(\Gamma^\infty)$-generic ...
12
votes
0
answers
191
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Are there times when replacement is "more natural" than collection?
There are a couple examples I'm aware of where choosing to axiomatize $\mathsf{ZF(C)}$ using collection instead of replacement results in a much nicer (or at least less surprising) picture:
Let $\...
5
votes
1
answer
163
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Can a generic ultrafilter over $\mathrm{NS}^+_{\omega_1}$ witness $\omega_1$ is Ramsey-like?
Suppose that $\kappa$ is an appropriate large cardinal (preferably a Woodin cardinal, but possibly something stronger) and let $G$ be a $\operatorname{Col}(\omega_1,<\kappa)$-generic filter over $V$...
5
votes
0
answers
200
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Are there Dedekind-infinite amorphous sets?
An amorphous set is an infinite set (i.e. cannot be put into bijection with any finite set $\{ 1, \dots, n \}$ for any $n$) that cannot be partitioned into two mutually disjoint infinite subsets. ...
14
votes
1
answer
570
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Changing the cofinality of a regular cardinal without collapsing any cardinals?
I have a short but hopefully interesting question on cardinal arithmetic and collapsing cardinals:
Is it possible to change the cofinality of a regular cardinal without collapsing any cardinals?
Is ...
2
votes
0
answers
113
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Adding partitions of one but not the other kind
Say that two partitions $(P_i)_{i\in I}, (Q_j)_{j\in J}$ are isomorphic iff there is a bijection $f: I\rightarrow J$ such that $\vert P_i\vert=\vert Q_{f(i)}\vert$ for all $i\in I$. (Note that in the ...
1
vote
0
answers
62
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Can the proper/whole domain relationship in bi-interpretations be reversed for non-synonymous theories?
Suppose we have theories $T$ and $H$ that are bi-interpretable, now suppose that the relevant interpretations achieving that bi-interpretability are: $\tau: T \to H$, and $\pi: H \to T$. Now suppose ...
9
votes
2
answers
1k
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Truth in a different universe of sets?
I understand that provability and truth as different concepts.
Provability is syntactic, it only concerns whether the given
sentence can be derived by reiterating the inference rules over a
collection ...
10
votes
1
answer
348
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1970 question of Reinhardt - how large is this ordinal?
On page 241 of William Reinhardt's paper "Ackermann's set theory equals ZF" (Annals of Math. Logic vol. 2, 1970), question 4.15 is the following:
How large is the first ordinal $\gamma$ ...
4
votes
0
answers
133
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Proof of: No rapid filter is Lebesgue measurable
I'm studying the following theorem in (Schindler, 2014: Set Theory
Exploring Independence and Truth), p. 178-180:
Theorem 9.16 (Mokobodzki) No rapid filter F $\subset$ ${}^\omega 2$ is Lebesgue ...
11
votes
2
answers
780
views
Undefinable inner model
What are some examples of a pair $M\subseteq N$ of transitive set models of $\mathsf{ZFC}$ with the same ordinals, such that $M$ is not a definable class (with parameters) in $N$? Is it possible that $...
0
votes
1
answer
212
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Is it consistent that $2^{(\cdot)}$ is "surjective" on the class of uncountable ordinals?
$\newcommand{\Z}{{\sf (ZFC)}}$
It is consistent in $\Z$ that there is an uncountable cardinal $\kappa$ such that for no cardinal $\lambda$ we have $2^\lambda = \kappa$: Take any model in which $2^{\...
4
votes
1
answer
456
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Truth Values of Statements in non-standard models
Excuse me, if the question sounds too naive.
Non-standard models of PA will have statements of non-standard lengths, basically infinite. And it is also true that every statement of a theory will have ...
2
votes
0
answers
142
views
Is it consistent to have these kinds of acyclic hereditarily size sets?
Working in $\sf ZFC-Reg. {}+ Acyclicity$. Where :
Acyclicity: $\neg \exists x_1, \cdots, \exists x_n: x_1 \in x_2 \in\cdots\in x_n \in x_1$
We add the following kind of weird non-well founded sets.
$\...
3
votes
1
answer
171
views
Is the universe of ZFA rigid? The pairing axiom implies that even atoms have a unitary set which discern them from all other atoms. So, is it rigid?
Although any permutation of atoms induces an automorphism of the whole universe, atoms seem to be indiscernible only within the permutation models. Can a permutation model be extended to a rigid ...
3
votes
0
answers
135
views
Lindström's theorem part 2 for non-relativizing logics
By "logic" I mean the definition gotten by removing the relativization property from "regular logic" — see e.g. Ebbinghaus/Flum/Thomas — and adding the condition that for every ...
4
votes
1
answer
156
views
Which of the known variants of Replacement can survive DeExtensionality?
Starting with $\sf ZF$. If we replace the power set axiom by the axiom stating that for any set $A$ there exists a set $x$ such that for every $y \subseteq A$ we have a set $y' \in x$ such that $\...
13
votes
1
answer
912
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Cantor-Bernstein with "weakly injective" functions
Let us call a map $f: X \to Y$ between non-empty sets a "weak injection" if $f^{-1}(\{y\})\subseteq X$ is finite for every $y \in Y$.
Recall that the (Schroeder-)Cantor-Bernstein-Theorem (...
5
votes
2
answers
403
views
Models of second-order arithmetic closed under relative constructibility
I know little to nothing about second-order arithmetic and its subsystems. However, I would like to understand when a model of (a subsystem of) second-order arithmetic ($\mathsf{Z}_2$) is downward ...
5
votes
0
answers
187
views
Reference-Request: Had this replacement principle been investigated before?
Replacement$^*$: If $\varphi$ is a formula with at least two variables occuring free, in which $``x,y"$ do not occur free, then:
$$\forall a \forall b \forall c \forall d \ ( \varphi(a,b) \land \...
2
votes
0
answers
124
views
The strongest reflection principle that does not violate covering lemmas
#-generated reflection, or Indiscernible-generation, is considered to be the strongest reflection principle that does not violate the covering lemma in L. [1]
Is there a way to extend this success to ...
7
votes
1
answer
228
views
Can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$?
Since $\operatorname{cf}(\aleph_\omega)=\omega$, $\aleph_\omega<\aleph_\omega^\omega$. However, can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$? I am especially interested in models for ...
3
votes
2
answers
120
views
Set sizes in linear set systems on $\mathbb{N}$ containing some disjoint sets
Is there a set $E\subseteq {\cal P}(\mathbb{N})$ of subsets of $\mathbb{N}$ with the following properties?
$|e| > 2$ for all $e\in E$,
$e_1\neq e_2 \in E$ implies $|e_1 \cap e_2| \leq 1$,
for all $...
2
votes
0
answers
95
views
Which of these non-well founded set theories is synonymous with ZFC?
Lets add a constant $\mathcal A$ to the language of $\sf ZFC$.
Let "Foundation$_{\mathcal A}$" denote the following sentence:
$$ \forall x: \forall y \in x \exists z \in y \cap x \to \exists ...
6
votes
1
answer
138
views
Preservation of cardinals implies preservation of cofinalities when $V=L$?
Kunen mentions the result stated in the title. It would be much appreciated if someone can give a reference for this. Thank you!
3
votes
0
answers
100
views
Existence of a almost increase $\omega_1^{\omega_1}$ sequence mod $[\omega_1]^{<\omega_1}$ with length $\omega_2$
In my textbook, the author said that the sequence below is satisfied the requirement.
$$\text{For }\alpha<\omega_1,\forall\gamma<\omega_1,g_\alpha(\gamma)=\alpha, \text{For }\omega_1\le\alpha<...
1
vote
0
answers
180
views
Can this Mereological system be synonymous with $\sf ZF(C)$?
This question is about synonymy of $\sf ZFC$ set theory with the following Mereological theory:
Language: first order logic with equality. Extra-logical primitives: $\subseteq$ standing for the binary ...
8
votes
1
answer
1k
views
Worst of both worlds?
It's well known that $\mathsf{AC}$ implies the existence of non-measurable sets. And it's also true that, if all sets are measurable, then $|\mathbb{R}/\mathbb{Q}| > |\mathbb{R}|$. But is there a ...
8
votes
1
answer
811
views
What is the least inaccessible cardinal for Tarski-Grothendieck set theory?
Let ordinal $\alpha$ be the least ordinal such that $V_\alpha\models$ Tarski-Grothendieck set theory.
What position does $\alpha$ have in the hierarchy of inaccessible cardinals?
2
votes
0
answers
102
views
Existence of trees with height $\kappa$, every level has at most size $\lambda$ and has at least $\lambda^{+}$ maximal branches
Definitions A tree means a set-theoretic tree, that is a poset $(T,<)$ so that for each $x\in T$, the set $\{y\in T\mid y<x\}$ is well-ordered.
A $\kappa$-Kurepa tree is a tree of height $\kappa$...
1
vote
1
answer
583
views
Can this kind of Mereology be synonymous with Set Theory?
This question is about synonymy of Morse-Kelley set theory "$\sf MK$" with the following Mereological theory:
Language: first order logic with equality. Extra-logical primitives: $\subseteq$ ...
8
votes
0
answers
180
views
Reference request: choiceless cardinality quantifiers
There is a substantial literature on the logic of cardinality quantifiers. (E.g., the quantifier $Q_\alpha$ where $M \vDash Q_\alpha x \, \varphi (x)$ iff $\vert \{a \in M : M \vDash \varphi[a] \} \...
4
votes
1
answer
282
views
Is ZFGC, minimally modified to allow two Quine atoms instead of the empty set, synonymous\bi-interpretable with ZFGC?
Working in $\sf ZFGC$, remove Foundation, stipulate the existence of exactly two Quine atoms. Restrict Separation to fulfillable formulas, i.e. $\{x \in A \mid \phi \}$ exists as long as $\phi$ holds ...
3
votes
2
answers
223
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Question regarding $W$ as not hyperarithmetic
Consider the indexes of all ordinary programs generating functions from $\mathbb{N}^2$ to $\{0,1\}$. If we let $W$ be the set of exactly of all those indexes $e$ such that $\phi_e$ computes a total ...
8
votes
0
answers
204
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The Hausdorff dimension of the set of reals of inner models
Suppose that both $M$ and $N$ are models of $ZFC$ with $M\subseteq N$ so that $M$ is definable in $N$.
Question Can $(\mathbb{R})^M$ have Hausdorff dimension strictly between $0$ and $1$ in $N$? How ...
-2
votes
2
answers
477
views
Can Mereology be bi-interpretable with Set Theory, in absence of the bottom object?
This question is about synonymy between Set theory and Mereology.
David Lewis in Mathematics is Megethology tried to reduce Set Theory to Mereology augmented with a singleton function. The following ...
12
votes
1
answer
446
views
Why do we need the comparison lemma?
An inner model is a standard transitive (proper class) structure which satisfies all the axioms of ZFC and contains all the ordinals. The simplest and most well-known inner model is Gödel’s $L$, which ...
6
votes
3
answers
779
views
Connected graphs isomorphic to their own contraction
Let $G = (V, E)$ be a simple, undirected graph with $|V|>2$, and let $S\subseteq V$ be a set with more than $1$ element. By $G/S$ we denote the graph obtained by collapsing $S$ to one point. More ...
7
votes
1
answer
217
views
Is the set of ordinals in Double Extension Set Theory really a set?
We got stuck on the definition of ordinals when we built the DEST(Double Extension Set Theory) checker on Cubical Agda and ...
-2
votes
1
answer
204
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Would this alteration safeguard the resulting theory from inconsistency?
If we replace "Emergence" axiom in the theory $T$ presented at posting "What is the set theory synonymous with this order-set theory?" with the following axiom, call the resulting ...
3
votes
1
answer
161
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Would this alteration of $T$ affect its synonymy with PA?
If we replace the single axiom "Emergence" in the axiomatic system $T$ presented at posting [What is the set theory synonymous with this order-set theory], by the following schema. Would the ...
3
votes
0
answers
173
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In constructive set theory, is it consistent for there to be a ring that models smooth infinitesimal analysis?
In a constructive set theory such as CZF, it is consistent to assume that every function $f : \mathbb R \to \mathbb R$ is continuous. However, it is not consistent to assume that every such function ...
1
vote
1
answer
296
views
What is the set theory synonymous with this order-set theory?
Let $ T$ be a theory written in Mono-sorted first order logic with equality, with extralogical primitives: $<, \in$.
Define: $x \leq y \iff x < y \lor x=y$
Axioms:
$\textbf{Well ordering: }\\\...
8
votes
2
answers
1k
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Follow up question: Shelah's "Can you take Solovay's inaccessible away?"
In this answer to the question " Shelah's "Can you take Solovay's inaccessible away?" " the following is stated:
Assume that $\aleph_1$ is not inaccessible in $L$, hence a ...
10
votes
0
answers
206
views
Is $\kappa \rightarrow [\kappa]^2_3$ the same as $\kappa \rightarrow [\kappa]^2_2$ for inaccessible $\kappa$
The principle $\kappa \rightarrow [\kappa]^2_\alpha$ states that whenever we have a coloring $c:[\kappa]^2\rightarrow \alpha$ there is $H \subset \kappa$ of size $\kappa$ s.t. $|c"[H]^2|<\alpha$.
...
1
vote
0
answers
80
views
Is every set equinumerous to a well founded set in acyclic ZF?
If we replace the axiom of Regularity in $\sf ZF$ by the scheme of Acyclicity, which is:
$$\begin{align} n=2,3,\dots;\ & \neg \exists x_1,\dots , \exists x_n: \\ &x_1 \in x_2 \land \dots \...