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.

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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 ...
Zoorado's user avatar
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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 ...
Dominic van der Zypen's user avatar
5 votes
1 answer
198 views

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 ...
Mohammad Golshani's user avatar
4 votes
1 answer
193 views

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 $\P(Y)$, ...
Dominic van der Zypen's user avatar
4 votes
1 answer
169 views

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 ...
Rupert's user avatar
  • 2,005
12 votes
0 answers
191 views

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 $\...
Noah Schweber's user avatar
5 votes
1 answer
163 views

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 $V$...
Hanul Jeon's user avatar
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5 votes
0 answers
200 views

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. ...
Ahraman's user avatar
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14 votes
1 answer
570 views

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 ...
user2925716's user avatar
2 votes
0 answers
113 views

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 ...
Noah Schweber's user avatar
1 vote
0 answers
62 views

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 ...
Zuhair Al-Johar's user avatar
9 votes
2 answers
1k views

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 ...
Student's user avatar
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10 votes
1 answer
348 views

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 $\gamma$ ...
C7X's user avatar
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4 votes
0 answers
133 views

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 ...
Caro Meier's user avatar
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 $...
new account's user avatar
0 votes
1 answer
212 views

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^{\...
Dominic van der Zypen's user avatar
4 votes
1 answer
456 views

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 ...
Amiren's user avatar
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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. $\...
Zuhair Al-Johar's user avatar
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 ...
Décio Krause's user avatar
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 ...
Noah Schweber's user avatar
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 $\...
Zuhair Al-Johar's user avatar
13 votes
1 answer
912 views

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 (...
Dominic van der Zypen's user avatar
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 ...
Lorenzo's user avatar
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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 \...
Zuhair Al-Johar's user avatar
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 ...
Ember Edison's user avatar
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 ...
Calliope Ryan-Smith's user avatar
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 $...
Dominic van der Zypen's user avatar
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 ...
Zuhair Al-Johar's user avatar
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!
LYS's user avatar
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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<...
X X's user avatar
  • 31
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 ...
Zuhair Al-Johar's user avatar
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 ...
Zemyla's user avatar
  • 309
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?
Frode Alfson Bjørdal's user avatar
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 $\kappa$...
George Marangelis's user avatar
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: $\subseteq$ ...
Zuhair Al-Johar's user avatar
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] \} \...
Beau Madison Mount's user avatar
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 ...
Zuhair Al-Johar's user avatar
3 votes
2 answers
223 views

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 ...
SSequence's user avatar
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8 votes
0 answers
204 views

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 ...
喻 良's user avatar
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-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 ...
Zuhair Al-Johar's user avatar
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 ...
Binary198's user avatar
  • 704
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 ...
Dominic van der Zypen's user avatar
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 ...
Ember Edison's user avatar
-2 votes
1 answer
204 views

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 ...
Zuhair Al-Johar's user avatar
3 votes
1 answer
161 views

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 ...
Zuhair Al-Johar's user avatar
3 votes
0 answers
173 views

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 ...
Christopher King's user avatar
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: }\\\...
Zuhair Al-Johar's user avatar
8 votes
2 answers
1k views

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 ...
C_M's user avatar
  • 83
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. $|c"[H]^2|<\alpha$. ...
Jiachen Yuan's user avatar
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 \...
Zuhair Al-Johar's user avatar