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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Systems of elementary embeddings

Recently I've been thinking about elementary embeddings, partition cardinals, etc. as part of my never-ending quest for understanding of consistency strength :p I came up with this idea, called I* ...
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-5 votes
0 answers
68 views

Are dark numbers known? [closed]

The limit in analysis is approximated better and better by a sequence or series. Contrary to set theory. Bertrand Russell wrote: "But nowadays the limit is defined quite differently, and the ...
1 vote
0 answers
49 views

Is "strongly unbounded logics are unbounded" equivalent to "no descending sequence of cardinals"?

This question is motivated by Vaananen's paper Generalized quantifiers in models of set theory. Say that a (set-sized, regular) logic $\mathcal{L}$ is unbounded if there are $\mathcal{L}$-sentences $\...
1 vote
1 answer
71 views

Are there known general tuple implementations that are 2 types high and withstand absence of extensionality and infinity?

Is there a general $\alpha$-tuple implementation that is of height $2$, that both doesn't require infinity of the naturals, and is at the same time stable under lack of Extensionality? My own try to ...
6 votes
0 answers
86 views

What large cardinals are needed to imply projective sets have the perfect set property?

If there are infinitely many Woodins, then every projective set is determined, whence every projective set has the perfect set property (PSP). Since determinacy is such a stronger property than the ...
2 votes
0 answers
106 views

Which cardinal $\kappa\geq \omega_1$ is critical for the following property...?

Which cardinal $\kappa\geq \omega_1$ is critical for the following property: Let $X\subset \mathbb R$ and $\kappa>|X|\geq \omega_1$. Then there is an uncountable family $\{X_{\alpha}\}$ such that $...
1 vote
0 answers
67 views

How to express Kunen's inconsistency, Reinhardt and Wholeness axioms, by single sentences?

Working in $\sf NBG, $ can we express the property of a class being set theoretically definable, by a single sentence? Like for example, the following way: $$\operatorname {std}(X) \iff \exists x_1 \...
18 votes
2 answers
810 views

Statements in differential geometry independent from ZFC

It is well known that some problems in functional analysis and in general topology are independent from ZFC: to name a few, Kaplansky's conjecture, the existence of outer automorphisms of the Calkin ...
0 votes
0 answers
150 views

On the definition of small categories in SGA4

We assume ZFC+U. A category is an ordered pair $(\operatorname{Ob} \mathcal{C},\operatorname{Mor} \mathcal{C},\operatorname{dom},\operatorname{codom},e,∘)$ of sets (not classes) and maps satifying ...
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3 votes
1 answer
182 views

Ramsey-like property with order condition

I wonder if there are regular cardinals $\lambda$ and $\kappa$ such that $\kappa < \lambda \leq 2^\kappa$ and such that, consistently, the following holds: Let $c: [\lambda]^2 \to \kappa$ be such ...
2 votes
0 answers
130 views

Can we have full choice prior to Reinhardt cardinals?

Working in $\sf ZF + Reinhardt \ cardinal$, can we have full choice over all stages $V_{\alpha < \kappa}$ where $\kappa$ is the Reinhardt cardinal, i.e., the critical point of the elementary ...
1 vote
0 answers
46 views

Can this method let choiceless large cardinals be smaller than cardinals compatible with choice?

Recall question "Can we have this sequence where choice fails and returns?" Can that theory be extended with requiring the $\mathcal V_n$'s to fulfill a choiceless large cardinal extension ...
4 votes
1 answer
175 views

Can we have this sequence where choice fails and returns?

Can we have a sequence of transitive sets $\langle\mathcal V_0, \mathcal V_1, \mathcal V_2,...\rangle$, all modeling $\sf ZF$, such that $\mathcal P(V_n) \subset \mathcal V_{n+1}$, and the cardinality ...
0 votes
0 answers
164 views

What is `the ideal set theory' [closed]

This Q hardly has much sense but anyway, is there anything called the ideal set theory known? Most assuredly this should have nothing to do with ideals (like the Frechet ideal etc.).
2 votes
0 answers
103 views

Is there a Lusin space $X$ such that ...?

Is there a Lusin space (in the sense Kunen) $X$ such that $X$ is Tychonoff; $X$ is a $\gamma$-space ? Note that if $X$ is metrizable and a $\gamma$-space then it is not Lusin. In mathematics, a ...
1 vote
0 answers
172 views

Harvey Friedman: The expanding mind

In reference 1, Friedman writes: I discuss my efforts concerning 3 crucial issues in the foundations of mathematics that are deeply connected with the great work of Kurt Gödel. [...] B. Are there ...
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2 votes
0 answers
186 views

Can we interpret Reinhardt cardinals this way?

To the language of set theory add a primitive unary predicate $\operatorname {Universe}$ and a primitive total unary function $j$. Add all axioms of $\sf ZF$ in the language of this theory, i.e. the ...
-1 votes
0 answers
77 views

Reinhardt cardinals in Paraconsistent Set Theory

The current developments in paraconsistent set theory have resolved Russell's paradox and maintained sufficient consistency strength, which leads to the following interesting questions. Is there a ...
6 votes
0 answers
121 views

Can we have a 'universal class' for elementary embeddings $j\colon V\to V$

Work over $\mathsf{GB}$, Gödel-Bernays set theory (without choice). My question is the following: Question. Is the following statement consistent with $\mathsf{GB}$? There is a universal class for ...
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1 vote
2 answers
223 views

Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?

Is it possible to iterate elementary embeddability and reflect on those stages that are elementary embeddable to themselves? The following is a formal capture of that idea: To the language of $\sf ZF$...
1 vote
1 answer
79 views

Recursively inaccessible ordinals and non locally countable ordinals

This answer seems to imply that: for an ordinal $\alpha$, to be recursively inaccessible (i.e. $\alpha$ is admissible and limit of admissible) implies to be not locally countable (i.e. $L_\alpha \...
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7 votes
1 answer
179 views

Does $\mathit{Aut}(\mathbb{R};+)$ have a copy in $L(\mathbb{R})$ granting large cardinals?

Throughout, work in $\mathsf{ZFC}$ + large cardinals (let's say a proper class of Woodin limits of Woodins but I'm happy to go higher if that would help). Let $\mathcal{R}=(\mathbb{R};+)$ be the ...
3 votes
1 answer
305 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: ...
0 votes
0 answers
153 views

Can acyclicity manage to elude Cantor's theorem?

A formula $\phi$ is acyclic if we can associate with its open expansion an non-directed acyclic graph, whose nodes are the terms of the formula and the edges are: if $\text{“}{\mathcal F}(v_1,\ldots,...
7 votes
1 answer
223 views

Complexity of infinitary satisfiability, part 2

This question is a follow-up to this one, which was almost entirely answered by Farmer S. Throughout, we work in $\mathsf{ZFC+V=L}$. Given a "pre-admissible" (= admissible or limit of ...
5 votes
1 answer
179 views

Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?

Recall that $\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the sfip which has no infinite pseudo-intersection $\}$. The cardinal $\mathfrak{q}_0$ defined as the smallest ...
-3 votes
0 answers
103 views

Is there a nontrivial elementary embedding $J \rightarrow WF$?

Preliminary assumptions. Each of the three elementhood relations, A ∈ B (stereotypically well-founded) A ∈ A (Quine-like) ... C ∈ B ∈ A (Mirimanoff-like) ... is assumed to correspond to either ...
8 votes
1 answer
840 views

Is there a form of choice that can elude Kunen's inconsistency theorem?

When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, ...
14 votes
1 answer
207 views

Is there a countably infinite closed interval in the lattice of topologies?

Is there an interval of the form $[\sigma,\tau]$ in the lattice of topologies on some set $X$ such that $|[\sigma,\tau]| = \aleph_0$? In other words, do there exist two topologies $\sigma$ and $\tau$ ...
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3 votes
1 answer
67 views

Extending almost disjoint family in a maximal set

Call a family of sets $\mathcal{F} \subseteq [\omega]^\omega$ maximal if there does not exist some $X \in [\omega]^\omega \setminus \mathcal{F}$ such that $X$ is almost disjoint with all elements of $\...
7 votes
3 answers
408 views

Elementary countable submodels in Gödel's universe

By the downward Lowenheim-Skölem theorem we can find two countable ordinals $\alpha < \beta$ such that $L_\alpha \prec L_{\omega_1}$ and $L_\beta \prec L_{\omega_1}$. That is, $L_\alpha$ and $L_\...
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14 votes
1 answer
478 views

How “disconnected” can a continuum be?

A continuum is a compact connected metrizable topological space. Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint ...
10 votes
1 answer
495 views

Universal property of the set of injections in the category of sets

Given two sets $A$ and $B$, the function set $B^A$ is characterized by the universal property that the functor $(-)^A:\mathrm{Set} \to \mathrm{Set}$ is the right adjoint of the functor $(-)\times A:\...
11 votes
1 answer
289 views

What is the "iterated definability" limit of first-order logic?

Roughly speaking, given a set-sized logic $\mathcal{L}$ let $\mathcal{L}'$ be gotten by adding to $\mathcal{L}$ the ability to quantify over $\mathcal{L}$-definable relations. (The details are a bit ...
6 votes
1 answer
790 views

Can the axiom of choice be proved with ZF+Tarski axiom?

Can choice be proved with ZF+Tarski axiom?
3 votes
0 answers
154 views

What is the name of the class of topological spaces with the following property ....?

What is the name of the class of topological spaces with the following property $P$ ? $X\in P$ iff for any open set $W$ in $X$ and any point $x\in \overline{W}\setminus W$ there is an open set $V$ ...
4 votes
1 answer
181 views

Is ordinal definability in terms of stages of cumulative size hierarchy equivalent to the usual one?

Working in $\sf ZF$ Define: $W_0 = \emptyset \\ W_{\alpha+1} = H_{\leq |W_\alpha|} =\{x \mid \forall y: y \in \operatorname {trcl} (\{x\}) \ |y| \leq |W_\alpha| \} \\ W_\lambda= \bigcup W_{\alpha < ...
17 votes
3 answers
822 views

Minimum transitive models and V=L

Is there a c.e. theory $T⊢\text{ZFC}$ in the language of set theory such that the minimum transitive model of $T$ exists but does not satisfy $V=L$? You may assume that ZFC has transitive models. ...
3 votes
1 answer
82 views

Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space?

A topological space $X$ is called a $\sigma$-space if every $F_{\sigma}$-subset of $X$ is $G_{\delta}$. A topological space $X$ is called a $Q$-space if any subset of $X$ is $F_{\sigma}$. Definition. ...
3 votes
1 answer
223 views

If we add stratified\acyclic replacement to the wholeness axiom, would that increase its consistency strength?

If we add to the wholeness axiom, the axiom of stratified\acyclic replacement, what would be the consistency state and strength of the resulting theory? The wholeness axiom $\sf WA$, introduced by ...
3 votes
1 answer
136 views

If a theory speaks of sets that cannot be forced to be parameter free definable, then does this entail a large cardinal property?

If we say that an effectively generated first order theory $\sf T$ extends $\sf ZF$, such that every countable model of $\sf T$ doesn't have a class forcing extension that is pointwise definable. ...
7 votes
1 answer
304 views

Can we have mutual elementary embeddability between distinct transitive sets?

Is it consistent with $\sf ZFC$ to have mutual elementary embeddability between distinct transitive sets? Formally, is the following theory consistent? $${\sf ZFC} + \exists M \exists N: M,N \text { ...
3 votes
1 answer
160 views

Definable closure in class-sized expansions of o-minimal groups

I am working in NBG set theory with limitation of size (i.e. the class of all sets is in bijection with the class of ordinals). Let $\mathbf{G}$ be a class-sized o-minimal expansion of an ordered ...
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1 vote
1 answer
131 views

Is there a model of each of the following kinds of theories in the first transitive model of ZFC?

The question of whether the minimal transitive model $M$ of $\sf ZFC$ have a model of each consistent extension $\sf T$ of $\sf ZF$, is answered to the negative, basically because $M$ is countable and ...
10 votes
2 answers
680 views

Does every consistent extension of ZF have a model in the minimal transitive model of ZFC?

Suppose there exists a transitive model of $\sf ZFC$. Is it the case that every consistent theory that extends $\sf ZF$ must have a model that is an element of the minimal transitive model of $\sf ZFC$...
1 vote
1 answer
74 views

Are externally pointwise definable models of ZFC subject to the same limitations of the internally pointwise definable ones?

By pointwise definable models, it's meant that every element of those models is definable after a formula in a parameter free manner, but that defining formula is in the language of that model, i.e., ...
6 votes
2 answers
159 views

Proper class of inaccessibles and 1-inaccessible cardinals

Hello I am starting to learn set theory and I am having some difficulty with these notions. Let's say we have a 1-inaccessible cardinal which means it is stronger than the assertion that there is a ...
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6 votes
1 answer
270 views

Which model is the minimal pointwise definable model of $\sf ZFC$?

Is the minimal transitive model of $\sf ZFC$ pointwise definable? If not, then what is the minimal pointwise definable model of $\sf ZFC$? Can we define that using Hamkins result for existence of ...
9 votes
0 answers
213 views

Feferman's universes for proof assistants?

This question was prompted by a discussion from another MO question about the consistency of ZFC. There are some mathematicians who are comfortable with ZFC but uneasy with large cardinals. For them,...
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2 votes
1 answer
95 views

Boolean prime ideal Theorem and the existence of nonmeasurable sets

In a well-written wikipedia article titled "Boolean Prime Ideal Theorem" (BPIT) the author states: "A not too well known application of the BPIT is the existence of non-measurable sets&...

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