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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What is the set of axioms for this theory extracted from what is provable in the minimal model of ZFC?

This post is a follow up of this one posted to MathStackExchange. Working in $\sf ZFC + \exists M \, (M \overset{trs} \models ZFC)$, lets define a theory $T_0$ as: $$(\varphi \ \epsilon \ T_0) \iff \...
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Set theory for category theory

Let $SC$ (sets and classes) denote the modification of Ackermann set theory laid out in F. A. Muller's Sets, Classes and Categories (denoted there by ARC -- a brief summary of the theory is given ...
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Strong form of $\mathtt{PSP}$ for $K_\sigma$ sets

Consider an uncountable perfect $K_\sigma$ set $X\subseteq \omega^\omega$, where $K_\sigma$ means countable union of compact sets, perfect means that $X$ has no isolated points and $\omega^\omega$ is ...
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Ordering uncountable set without axiom of choice [closed]

Is anyone interested in an algorithm that builds a well ordered, uncountable, subset of the real numbers (irrational > 1) without the axiom of choice? I think I have found such an algorithm, but am ...
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At which large cardinal, the theory of the minimal transitive model of ZFC starts proving its absence?

Let's take the minimal transitive model of $\sf ZFC$ which, I came to know, is some minimal $L_\kappa$ for a countable $\kappa$, that models $\sf ZFC$, and since its minimal so no subset of it can be ...
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What's the consistency strength of this kind of cardinal?

My friend introduced the following notion: Let $\xi > 0$, $\eta$ be ordinals, $n$ be a natural number and $\mathcal{A}, X$ be classes. A cardinal $\kappa$ is called $\mathcal{A}\textrm{-}\eta\...
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Does this proof by Shelah use any "hidden assumptions"?

Recall that the approachability ideal for $\kappa$, denoted $I[\kappa]$, consists of all sets $A\subseteq\kappa$ such that there is a sequence $\overline{a}=(a_{\alpha})_{\alpha\in\kappa}$ of bounded ...
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Necessary and sufficient conditions for a forcing to add a Cohen real

Are there some necessary and sufficient conditions for a forcing notion to add a Cohen real in the generic extension? In other words, given a non-atomic forcing p.o. $\mathbb{P}$, what are the ...
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Does a well-pointed topos with enough projectives satisfy the internal axiom of chioice?

If yes, then I am also wondering if being well-pointed can be weakened to boolean (i.e. this is in the context of using Set as our metalogic so that well pointed Topoi are boolean). If not, then any ...
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Is this strengthening of the definition of weakly Shelah cardinals equivalent to being weakly Shelah?

This MathOverflow question by Trevor Wilson defines weakly Shelah cardinals as follows: A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \kappa$ there is some $\alpha < \kappa$ that ...
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Efficiency of covers

Let $X\neq \emptyset$ be a set. We say $C \subseteq {\cal P}(X)$ is a cover of $X$ if $\bigcup C = X$. For covers $C, D$ of $X$ we say that $C$ is more efficient than $D$ if $|C\setminus D| < |D \...
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Reinhardt's ultimate classes

In the preface to Sets and Classes by Muller, several research programs are outlined that were in development concurrently with publication (or finished slightly beforehand) that he would have liked ...
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Why may club Berkeley cardinals not be Berkeley?

"Large Cardinals beyond Choice" makes the following definitions: $\delta$ is a Berkeley cardinal if for every transitive set $M$ such that $\delta \in m$ and every $\eta \lt \delta$ there ...
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What's the consistency strength of this strengthening of weakly superstrong cardinals?

Recall that a cardinal $\kappa$ is weakly superstrong if, for every $A \subseteq V_\kappa$, there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \...
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Are equinumerous size preserving models of a theory isomorphic?

If by a size preserving model we mean any bijection between any two elements of it is an element of it. Then: is it a thoerem of $\sf ZFC$ that for any theory $T$ any two equinumerous size preserving ...
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How often are forcing extensions of countable computably saturated models of $\mathsf{ZFC}$ computably saturated?

Recall that given a finite language $\mathcal{L}$, we say that an $\mathcal{L}$-structure is computably saturated (or recursively saturated) if for any computable set $\Sigma(\bar{x},y)$ of $\mathcal{...
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Can ZFC sets be interpreted as single rooted trees with accessible degree and countable height?

Let $T$ be a single directed tree, by parameters $(\kappa, \lambda, \zeta)$ of $T$ we mean: the number of root nodes in $T$, the strict upper bound on the number of children nodes per a node in $T$, ...
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14 votes
1 answer
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Can $\kappa^\lambda$ be large if $2^\lambda$ is small and $\lambda<\mathrm{cof}(\kappa)$?

We work in ZFC throughout. The following question was posed to me by a friend: Can there exist cardinals $\kappa,\lambda$ such that $\lambda<\mathrm{cof}(\kappa)$ and $2^\lambda<\kappa<\...
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3 votes
1 answer
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$\mathtt{PSP}$ holding only for sets of cardinality $\mathfrak{c}$

Consider the sentence $\mathtt{PSP}_\mathfrak{c}$: "Every subset of $\mathbb{R}$ having the cardinality of the continuum contains a Cantor set". A priori this sentence is weaker than the ...
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5 votes
1 answer
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A question about the axiom of dependent choice

Let $\mathrm{NBG}^-$ be $\mathrm{NBG}$ minus the Axiom of Choice for Classes (including sets)). Further let $\mathrm{DC}$ be the Axiom of Dependent Choice for sets and $\mathrm{DC}^\omega$ be Bernays ...
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3 votes
1 answer
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Do all limit $\alpha \in \omega_1^L$ satisfy $L_\alpha \models V=HC$?

In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal and the start of a gap are defined as follows $\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\bigcap \mathcal{P}(\...
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Thinning chains of elementary extensions

I'm bumping this question, since I'm still curious regarding the answer but this question seems to have gone unnoticed. This is a follow-up to this question, regarding a stronger variant of ...
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Why can't $L_\beta$ contain a real coding a well-ordering of order-type $\beta$, when $\beta$ is a gap ordinal?

In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal is defined as follows $\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\cap \mathcal{P}(\omega) = \emptyset$ Their ...
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How do chains of elementary extensions compare to shrewdness?

I thought of the following large cardinal axiom, extending the notion of $\theta$-upliftingness: Let $\eta$ be be an ordinal, and $X$ be a class of ordinals. $\kappa$ is called $\eta$-iteratively ...
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2 votes
1 answer
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What's the consistency status/strength of this limitation principle?

$\DeclareMathOperator\iCard{iCard}$In a prior posting If we limit matters what ZFC can prove, would that be consistent? to MO, I tried to capture the informal principle of whatever ZFC proves, it is, ...
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2 votes
2 answers
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When is the Minkowski sum of weighted compact sets $w_1 B_1 + w_2 B_2 + \ldots$ (with $w \in L^1$) closed?

Let $B_1,B_2,\ldots,$ be compact subsets of $\mathbb R^d$ and $w_1,w_2,\ldots$ be nonnegative numbers summing to $1$. Consider the set $$ A := w_1 B_1 + w_2 B_2 + \ldots = \left\{\sum_{n=1}^\infty w_n ...
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2 votes
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What is the consistency strength of the following pattern of failure of the continuum hypothesis?

What is the least theory in which the following sentence is proved? $ \exists M: M\text { is CTM(ZFC+ GCH)} \land \forall \kappa \in Card^M (\kappa > 1 \implies \\\exists N: N \text { is CTM(ZFC) }...
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4 votes
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A question regarding the Hahn-Banach theorem and Banach limits

Set theorists typically prove the existence of Banach limits (EBL) using the Ultrafilter Theorem or, its equivalent, the Boolean Prime Ideal Theorem (BPI). Analysts, on the other hand, typically prove ...
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5 votes
2 answers
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Properties of Jech's hierarchy of stationary sets (Exercise 8.13, 8.14 of Jech)

I must first preface that while this is indeed a question on an exercise, I believe this is advanced enough for MathOverflow. Let $\kappa$ be a regular uncountable cardinal. Recall that the notion of ...
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-1 votes
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Is the following relation between size-unreachability and fulfilling continuum hypothesis correct?

If we define $\operatorname {size-unreachable}$ set as a set of all subsets of it of strictly smaller cardinality than it. Formally, we define: $$ \operatorname {size-unreachable}(X) \iff X=\{Y \...
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-2 votes
1 answer
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Which extension of ZFC proves that ZFC can only prove CH satisfied by the first two sets?

Which extension of $\sf ZFC$ prove that $$ {\sf ZFC} \not \vdash \exists x \, ( \operatorname {CH}(x) \land x \neq \emptyset \land x \neq 1)$$ Where $\operatorname {CH}(x) \iff \neg \exists \kappa \, (...
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-2 votes
1 answer
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If we limit matters what ZFC can prove, would that be consistent?

I was thinking about a principle that occurred to me regarding provability in ZFC and truth. The principle outrageously states that: whatever ZFC shows, it is! In other words whatever ZFC can prove ...
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3 votes
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Forcings that preserve $\mathtt{PSP}$

By $\mathtt{PSP}$ I mean the statement "every subset of the reals has the perfect set property, i.e. either is countable or it contains an homeomorphic copy of the Cantor space $2^\omega$". ...
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3 votes
1 answer
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When does the cardinality of a set equal the cardinality of an element of $V_\lambda$ for $\lambda$ being a limit ordinal?

Consider the following proposition. Proposition: let $\lambda$ be a limit ordinal and $V$ be the cumulative hierarchy starting with the null set, and $S$ be a set with $\vert S\vert<\vert V_\...
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3 votes
1 answer
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Equivalences of $\mathcal{F}$-Mahloness

Taken from Math Stack Exchange. Let $\mathcal{F}$ be a set of $\mathcal{L}_\in$-formulae, $\kappa$ be a cardinal and $A \subset \textrm{Ord}$. Then, $\kappa$ is called $\mathcal{F}$-Mahlo if $A \cap \...
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Are those two theories equivalent?

Lets denote any set that is "the set of all strictly smaller subsets of it" as size- unreachable. Formally: $$\operatorname {size-unreachable}(X) \iff \\\forall Y (Y \in X \iff Y \subset X \...
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Is definable bounded separation equivalent to bounded separation?

If we have all axioms of Mac Lane set theory except Separation and add to them the schema of definable bounded separation, then would the resulting theory be equivalent to Mac Lane set theory? ...
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5 votes
1 answer
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Shelah's "Can you take Solovay's inaccessible away?"

I was wandering if there was a book, thesis or some notes where Shelah's argument for $\mathtt{ZF}+\mathtt{DC}+$"All sets of reals are Lebesgue measurable" is equiconsistent with $\mathtt{...
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3 votes
1 answer
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Existence of a non-$Q$-set without the perfect set property

We have the following theorem: Suppose $\omega_1^L=\omega_1$ then there exists a $\Pi_1^1$ subset of reals without the perfect set property Moreover, under the same hypotheses, we can prove actually ...
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3 votes
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Co-analytic $Q$-sets

A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have ...
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1 vote
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Is the beth function continuous without the axiom of choice?

Suppose the beth function is defined as follows: beth[a]=|V[a]|, for all ordinals a. Here V[a] is the ath level of the cumulative hierarchy, and || is the cardinality function defined as in for ...
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4 votes
1 answer
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$\mathtt{PSP}$ implies the consistency of inaccessible cardinals

I'm looking for the proof that $\mathtt{PSP}$, the statement that every uncountable subset of the the Baire space $\mathbb{N}^\mathbb{N}$ contains an homeomorphic copy of the Cantor space $2^\mathbb{N}...
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What is the strength of allowing multiple predecessor numbers?

If we have a theory of numbers, pairs of numbers, and sets of those, and axiomatize that the relation $<$ on numbers is both extensional and well founded, then this theory would prove all PA axioms ...
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5 votes
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What is the most "Icarus" Icarus set axiom?

We call a set $X ⊆ V_{λ+1}$ an Icarus set if there is an elementary embedding $j : L(X, V_{λ+1}) ≺ L(X, V_{λ+1})$ with $\mathrm{crit}(j)< λ$. But this raises the question: What is the most "...
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3 votes
1 answer
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Why does the second smallest worldly cardinal believe the smallest worldly cardinal is worldly?

It is known that the property of being a worldly cardinal is not absolute (a cardinal $\kappa$ is worldly iff $V_{\kappa} \vDash \textsf{ZFC}$). See here and here for more. This said, it is the case ...
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4 votes
1 answer
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Does the partition principle imply (DC)?

For sets $x, y$ we write $x\leq y$, if there is an injection $\iota: x \to y$, and we write $x \leq^* y$ if either $x = \emptyset$ or there is a surjection $s: y \to x$. In ${\sf (ZF)}$ we have that $...
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6 votes
0 answers
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Fragments of infinitary logic with a weak definability property

For a countable admissible ordinal $\alpha$, let $\mathcal{L}_\alpha=\mathcal{L}_{\infty,\omega}\cap L_\alpha$ and let $\equiv_\alpha$ be the corresponding elementary equivalence relation. Say that ...
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1 vote
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Can Reinhardt cardinals be compatible with Choice in absence of Extensionality?

Is the proof of existence of Reinhardt (and higher) cardinals violating Choice dependent on Extensionality in an essential manner? What I mean is if we work in $\sf ZFA$ would it be possible to have a ...
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1 vote
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Supercompact cardinal above a measurable and fixed points of the ultrapower map

Let $\kappa$ be a measurable cardinal and let $j:V\to M$ be the ultrapower map. Assume $\mu$ is a supercompact cardinal with $\mu>\kappa$. What can we say about $j(\mu)$? Is it true that $j(\mu)=\...
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2 votes
1 answer
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Images of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$

My question is: Is every Polish space image of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$? Where a Polish space is a separable and completely metrizable space and where $\Bbb{N}^\...
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