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.

1,087 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
9 votes
0 answers
242 views

Distributivity of certain infinite products

Suppose we have a sequence of posets $\{\mathbb P_n : n\in\omega\}$ such that for each $n$, $\mathbb P_{n+1}$ is $|\mathbb P_n|^+$-distributive. Is $\prod_{n>0} \mathbb P_n$ necessarily $|\mathbb ...
Monroe Eskew's user avatar
  • 18.1k
9 votes
0 answers
160 views

Strong Chains in $^{\omega_1}\omega_1$ of length $\omega_3$

In a previous question, I asked about the impact of strong chains in $^{\omega_1}\omega_1$ (e.g., sequences of functions $\langle f_\alpha:\alpha<\kappa\rangle$ in $^{\omega_1}\omega_1$ that are ...
Todd Eisworth's user avatar
9 votes
0 answers
249 views

$\Sigma^2_2$ absoluteness and $\diamondsuit$

This question touches on recent questions concerning the role of $\diamondsuit$ and the Continuum Hypothesis. In particular, I would like to know more information about a conjecture of Woodin on the ...
Todd Eisworth's user avatar
9 votes
0 answers
371 views

On the role of $\diamondsuit$

The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets with the property that for any $A\...
Todd Eisworth's user avatar
9 votes
0 answers
355 views

A $\mathsf{ZF}$ example of two Baire spaces whose product is not Baire?

Motivated by this question I'm looking for a pair of Baire topological spaces whose product is not Baire and whose construction does not need the axiom of choice. The example of such spaces I'm ...
Alessandro Codenotti's user avatar
9 votes
0 answers
311 views

Uncountable group with no proper subgroup of maximal cardinal

The Prüfer group $\mathbb{Z}(p^\infty)$, for $p$ prime, has the interesting property that it is infinite, and every proper subgroup is finite, but can be arbitrarily large, so there is no proper ...
Dominic van der Zypen's user avatar
9 votes
0 answers
221 views

Does there exist a non-trivial elementary embedding from an ultrapower $V^{I}/U$ to $V^{I}/U$?

Does there exist a set $I$ and an ultrafilter $U$ on $I$ and a non-trivial elementary embedding $j:V^{I}/U\rightarrow V^{I}/U$? So the Kunen inconsistency result states that there does not exist a ...
Joseph Van Name's user avatar
9 votes
0 answers
360 views

Reflection principle for intuitionistic Zermelo–Fraenkel?

The well-known reflection principle for classical Zermelo–Fraenkel states: For any formula $\varphi(x_1,\ldots,x_n)$ of the language of ZFC with free variables $x_1,\ldots,x_n$, ZFC proves $$ \...
Ingo Blechschmidt's user avatar
9 votes
0 answers
225 views

Sacks property for higher cardinals

It is well known, that the Sacks forcing has the property that for any $f: \omega \rightarrow \omega$ in generic extension, one can obtain $F:\omega \rightarrow [\omega]^{<\omega}$ in ground model, ...
Ziemowit Kostana's user avatar
9 votes
0 answers
189 views

Analogue of strong stationary reflection from MM

Foreman-Magidor-Shelah proved that if Martin’s Maximum holds, then for every regular $\kappa>\omega_1$, every stationary $S \subseteq S^\kappa_\omega$ contains a club of ordertype $\omega_1$. This ...
Monroe Eskew's user avatar
  • 18.1k
9 votes
0 answers
290 views

ZF + "every Suslin set of reals is ${\bf \Sigma}^1_2$"

What is known about the theory ($\ast$) ZF + "every Suslin set of reals is ${\bf \Sigma}^1_2$"? By "reals" I mean elements of the Baire space $\omega^\omega$. For a cardinal $\kappa$, a set of ...
Trevor Wilson's user avatar
9 votes
0 answers
254 views

How many iterations of inner models/generic extensions are sufficient?

Let $M=M_0$ be a ctm of ZF. If $(M_0, \ldots, M_n)$ is a sequence of ctms of ZF, where for all $i,$ either $M_{i+1}$ is an inner model of $M_i$ or a generic extension of $M_i,$ then call $M_n$ an $n$-...
Elliot Glazer's user avatar
9 votes
0 answers
253 views

A bi-modal logic related to determinacy

The short version of my question is as follows. There is a natural (I hope!) way to associate a bimodal theory to a game (two-player, perfect-information, length-$\omega$, on $\omega$); are there "...
Noah Schweber's user avatar
9 votes
0 answers
347 views

Direct limits of $\sigma$-centered forcing notions

It is quite well known that Any FS (finite support) iteration of length $<\mathfrak{c}^+$ of $\sigma$-centered posets is $\sigma$-centered (see e.g. here). Now consider the following question: ...
dragoon's user avatar
  • 763
9 votes
0 answers
295 views

Moschovakis' discovery of E-recursion

E-recursion is a notion of generalized computability theory which seeks to extend computations to allow arbitrary sets as inputs. In contrast with e.g. $\alpha$-recursion, it disallows unbounded ...
Noah Schweber's user avatar
9 votes
0 answers
210 views

What large cardinal axioms does the point of first difference between elementary embeddings satisfy?

Let $j,k:V_{\lambda}\rightarrow V_{\lambda}$ be inequivalent elementary embeddings. Then let $\theta(j,k)$ be the largest limit ordinal $\gamma$ such that $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for ...
Joseph Van Name's user avatar
9 votes
0 answers
293 views

On an unpublished result of Magidor

In 1970th, Magidor proved the following important results: (1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$ is strong limit and $2^{\aleph_\omega}=\aleph_{\...
Mohammad Golshani's user avatar
9 votes
0 answers
321 views

preserving saturated ideals

A reliable source made the following claim: Suppose CH there is an $\omega_2$-saturated ideal on $\omega_1$. Then this is preserved by $\mathrm{Add}(\omega_1,\omega_2)$. Question 1: How do you ...
Monroe Eskew's user avatar
  • 18.1k
9 votes
0 answers
273 views

Proving regularity properties from forcing axioms

It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc. Is there a direct proof ...
Haim's user avatar
  • 421
9 votes
0 answers
173 views

Is it possible that all ultrafilters are determined by the meet-semilattice of sub-ultrapowers?

Suppose that $\mathcal{Z}$ is a filter on a set $X$. Let $\Pi(X)$ denote the lattice of all partitions of the set $X$. Then $(\Pi(X),\wedge)$ is a meet-semilattice where $P\wedge Q=\{R\cap S|R\in P,S\...
Joseph Van Name's user avatar
9 votes
0 answers
239 views

countable OD sets in the Solovay model

It is known that the following is true in the Solovay model (SM) for ZFC: any countable OD (ordinal-definable) set $X$ of reals necessarily consists of OD elements. What about countable OD sets of ...
Vladimir Kanovei's user avatar
9 votes
0 answers
599 views

Two questions about universally measurable sets

I have two questions about universally measurable sets: (1) Is there a universally measurable set of reals which does not have the Baire property? (2) Is there a universally measurable set of reals ...
Ashutosh's user avatar
  • 9,771
9 votes
0 answers
265 views

Which forcing types preserve the axiom of determinacy?

Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy? To be more specific, in Which forcings ...
Asaf Karagila's user avatar
  • 38.1k
9 votes
0 answers
289 views

Co-Heyting Valued Models of Paraconsistent Set Theory

I've been trying to do some forcing arguments in intuitionistic ZF using Heyting valued models where the Heyting algebra I'm using is actually a bi-Heyting algebra (both a Heyting algebra and a co-...
King Kong's user avatar
  • 631
9 votes
0 answers
523 views

"Hard" separation results in reverse mathematics (or similar)

This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...
Noah Schweber's user avatar
9 votes
0 answers
247 views

Consistency strength of saturated ideals on $[\lambda]^{<\kappa}$

In the Handbook of Set Theory, Foreman notes that in models constructed by Magidor from a huge cardinal, there exists a normal ideal on $[\omega_2]^{<\omega_1}$ that is $\omega_3$-saturated. Other ...
Monroe Eskew's user avatar
  • 18.1k
9 votes
0 answers
335 views

The global dimension of fields

In the absence of the Axiom of Choice, it is not necessarily true that all vector spaces over a field have bases. What are the possible global dimensions of fields in a model of ZF in which AoC ...
Mariano Suárez-Álvarez's user avatar
9 votes
0 answers
357 views

Well-founded families of sets and topological convergence

Background/Motivation A space is scattered if every non-empty subset has an isolated point. A space is pseudoradial if every non-closed set contains a transfinite sequence (a well-ordered net) ...
Santi Spadaro's user avatar
9 votes
0 answers
396 views

Can second order arithmetic make $\aleph_1^L$ countable?

Simpson's book Subsystems of Second Order Arithmetic shows $Z_2$ can interpret some fragments of ZF strong enough to give good theories of constructible sets and formalize statements like "there is a ...
Colin McLarty's user avatar
9 votes
1 answer
843 views

$\Pi_0^1$-weakly indescribable cardinals are exactly the regulars

Hi, I'm not sure if I should ask here or over at math.stackexchange.com, but I think here it's a bit more fitting. This question stems from a homework problem: Definition: Given some class of formulas ...
Apostolos's user avatar
  • 341
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
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
  • 4,191
8 votes
0 answers
422 views

What is the nicest bijection $\textbf{R}^p \to \textbf{R}^q$ that you know?

It is well-known that bijection between $\textbf{R}^p$ and $\textbf{R}$ exist (e.g. here, though many other examples exist). The problem with all these examples of bijections is that typically the ...
8 votes
0 answers
184 views

Can every Borel set be partitioned into $\leq\!\aleph_1$ $F_{\sigma \delta}$ sets?

Consider the following two facts, a modified version of which appear in this paper of Arnie Miller from the early 1980's: $\bullet$ If $\mathbb R$ can be partitioned into $\aleph_1$ closed sets, then ...
Will Brian's user avatar
  • 17.4k
8 votes
0 answers
171 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 ...
Kameryn Williams's user avatar
8 votes
0 answers
162 views

Upper-bounding determinacy

While the converse of Borel determinacy ("If a set of reals is determined, then it is Borel") is boringly disprovable, I'm curious if there is a sense in which something like it is ...
Noah Schweber's user avatar
8 votes
0 answers
164 views

Does determinacy imply unravellability for the Borel sets (over a weak base theory)?

As far as I know, the only way we currently know how to prove Borel determinacy in $\mathsf{ZFC}$ is to go through unravelability (a rather technical property whose definition can be found in Martin's ...
Noah Schweber's user avatar
8 votes
0 answers
219 views

Absoluteness of the core model under a proper class of completely Jónsson cardinals

Example 2.4.2 of Larson's The stationary tower (based on Woodin's lecture) describes how we can absorb a generic filter over a set forcing in a definable inner model into a generic extension by $\...
Hanul Jeon's user avatar
  • 2,774
8 votes
0 answers
169 views

Intuition for branch uniqueness in inner model theory

In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage? At the level ...
Dmytro Taranovsky's user avatar
8 votes
0 answers
494 views

Ring of global sections of a functor $\mathbf{CRing} \to \mathbf{Set}$

Let $U : \mathbf{CRing} \to \mathbf{Set}$ be the forgetful functor. For any functor $F : \mathbf{CRing} \to \mathbf{Set}$ consider the class of natural transformations $$\mathcal{O}(F) := \mathrm{Hom}(...
Martin Brandenburg's user avatar
8 votes
0 answers
276 views

Inner models from highly saturated ideals

Solovay proved that a measurable cardinal is equiconsistent with the existence of an extension of Lebesgue measure to a full measure on $P(\mathbb R)$. The inner model direction is relatively simple. ...
Monroe Eskew's user avatar
  • 18.1k
8 votes
0 answers
411 views

Choice function for elementary embedding $j: V_\lambda\prec V_\lambda$

Work in ZF. Let $\lambda$ be strongly inaccessible. Let $\kappa$ be such that for every $\alpha\lt\lambda$, there is some $j: V_\lambda\prec V_\lambda$, with critical point $\kappa$, such that $j(\...
Master's user avatar
  • 1,103
8 votes
0 answers
252 views

Are there analogues of real-valued measurability for larger powersets?

Apologies if the question is somewhat naïve - I'm not a set theorist, just an enthusiast. One possible intuition we could have about the generalized continuum hypothesis is that power sets are so ...
zeb's user avatar
  • 8,533
8 votes
0 answers
307 views

A $\mathsf{ZF}$ example of a nonreflexive group which is isomorphic to its double dual?

Given a group $G$ denote by $G^\ast=\mathrm{Hom}(G,\Bbb Z)$ its dual and by $j\colon G\to G^{\ast\ast}$ the canonical homomorphism $g\mapsto (f\mapsto f(g))$. A group is reflexive iff $j$ is an ...
Alessandro Codenotti's user avatar
8 votes
0 answers
362 views

Can the set of endomorphisms of $(\mathbb R,+)$ have cardinality strictly between $\frak c$ and $\frak{c^c}$?

Let $\frak c$ be the cardinality of the reals. I know that in ZF the set of endomorphisms of $(\mathbb R,+)$ can have at least two different cardinalites: If we allow the axiom of choice, you can ...
user avatar
8 votes
0 answers
203 views

ladder system uniformization at successors of singulars

Shelah proved (paper 667) that if GCH holds and $\lambda$ is singular, then for every stationary $S \subseteq \{ \alpha < \lambda^+ : \text{cf}(\alpha) = \text{cf}(\lambda) \}$, there is a ladder ...
Monroe Eskew's user avatar
  • 18.1k
8 votes
0 answers
538 views

Does second order ZFC conservatively extend first order ZFC?

If I replace the axiom schema of specification in ZFC by a single axiom in second order logic, and similarly do same thing for the axiom schema of replacement, is this version of "second order ZFC" ...
user9730905's user avatar
8 votes
0 answers
1k views

Can every set be measurable?

The Solovay model shows that ZF (plus an inaccessible) is consistent with every subset of $\mathbb{R}$ being measurable. How far can we go in that direction? We can always have non-measurable sets ...
arsmath's user avatar
  • 6,720
8 votes
0 answers
1k views

Would Ultimate L really "[reduce] all questions of set theory to axioms of strong infinity"?

According to these slides, the axiom $V = \mathrm{Ultimate} \,L$ has the following consequences (p. 55): It implies the Continuum Hypothesis. It reduces all questions of set theory to axioms of ...
goblin GONE's user avatar
  • 3,693
8 votes
0 answers
224 views

Metrically Ramsey ultrafilters

On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the ...
Taras Banakh's user avatar
  • 40.8k

1 2 3
4
5
22