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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.

6
votes
0answers
388 views

Is there any theorem achieving Conway's “Mathematician's Liberation Movement”

John Conway on in the appendix to part zero of ONAG describes a "Mathematician's Liberation Movement". The goal would be to give mathematicians the freedom to create mathematical theories with the ...
14
votes
0answers
439 views

Does inner model theory seek canonical models for large cardinals?

Like the author of this question, I have heard that a main goal of inner model theory is building canonical inner models for large cardinals. My questions are: (a) Is this accurate? (b) If so, in ...
1
vote
0answers
57 views

Which varieties are compatible with the classical Laver tables?

Let $$A_{n}=(\{1,\dots,2^{n}-1,2^{n}\},*_{n})$$ denote the $n$-th classical Laver table. The operation $*_{n}$ is the unique binary operation on $\{1,\dots,2^{n}\}$ such that $$x*_{n}(y*_{n}z)=(x*_{n}...
5
votes
0answers
72 views

The descriptive complexity and definiteness of the space of all elementary embeddings $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$

Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Suppose that $(\alpha_{n})_{n}$ is an increasing cofinal sequence in $\lambda$. Give $\...
3
votes
0answers
58 views

Can a finitely generated algebra of rank-into-rank embeddings grow at rate $O(n\cdot\log(n))$?

Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $j\in\mathcal{E}_{\lambda}$ is a non-trivial elementary embedding, then define $\mathrm{crit}...
6
votes
1answer
250 views

Covering compact Hausdorff spaces with closed $G_\delta$ sets

I'm thinking about results of the form: Under assumption $A$, if $X$ is a compact Hausdorff space and $C$ is a cover of $X$ by closed $G_\delta$ sets, then there is a subcover of cardinality $\leq\...
6
votes
2answers
300 views

Poset dimension and width (Dilworth's theorem)

For a given poset $P$, let $\mathrm{dim}(P)$ denote the least cardinal $\kappa$ such that there exists a $\kappa$-sized collection of linear extensions of $P$, say $\mathcal{L}$, such that $\leq_P = \...
4
votes
1answer
271 views

Stronger negation of AC given by rejecting “infinite hat” puzzles

Some of the strangest implications of AC are the "infinite hat" puzzles, which are on Wikipedia, and have been talked about on MO several times, including some variants. There are different ways to ...
3
votes
1answer
239 views

Hartogs' Number of the Reals and $\Theta$ without choice

There are two important numbers that in some meaningful sense describe "how well-orderable" the reals are: Hartogs' Number $H(\Bbb R)$, also notated as $\aleph(\Bbb R)$, the least ordinal/well-...
10
votes
4answers
1k views

When is it okay to intersect infinite families of proper classes?

For experts who work in ZFC, it is common knowledge that one cannot in general define a countable intersection/union of proper classes. However, in my work as a ring theorist I intersect infinite ...
0
votes
0answers
105 views

Strength of $Δ^1_{2n}$ determinacy

According to Lightface mice with finitely many Woodin cardinals from optimal determinacy hypotheses by Yizheng Zhu, theorem 1.1, over $\mathbf{Σ^1_{2n+1}}$ determinacy, $Δ^1_{2n+2}$ determinacy is ...
4
votes
1answer
170 views

“Surjective cardinals” - using surjections rather than injections to define isomorphism classes of sets

Cantor used the notion of an "injection" to formalize the size of two sets: A is "smaller" than B if A injects into B. Simply put, the question is - how does this situation change if we use ...
0
votes
0answers
106 views

Can Laver tables go extinct?

An algebra $(X,*)$ is said to be self-distributive if it satisfies the identity $x*(y*z)=(x*y)*(x*z)$ for all $x,y,z\in X$. If $(X,*)$ is an algebra, then a subset $L\subseteq X$ is said to be a left-...
2
votes
1answer
45 views

Hypergraph colorings with small fibers

Let $H=(V,E)$ be a hypergraph such that for every $e\in E$ we have $|e|\geq 2$. A map $c:V\to \kappa$, where $\kappa$ is a cardinal, is said to be a (hypergraph) coloring if for all $e\in E$ the ...
2
votes
0answers
63 views

Is there a normal separable sequential $\aleph$-space with uncountable extent?

It is a classical fact from the undergraduate course of General Topology that under CH (more precisely, under $2^{\omega_1}>\mathfrak c$) every separable normal space has countable extent, i.e., ...
0
votes
1answer
77 views

Size of edge set of infinite hypergraphs with $\chi(H) = |V(G)|$

Let $H=(V,E)$ be a hypergraph such that for every $e\in E$ we have $|e|\geq 2$. A map $c:V\to \kappa$, where $\kappa$ is a cardinal, is said to be a (hypergraph) coloring if for all $e\in E$ the ...
2
votes
0answers
69 views

Does the period of the first row in the odd size bad Laver tables grow without bound?

Does the length of the period of the first row in the odd bad laver tables grow without bound? If $n$ is a natural number, then the $n$-th bad Laver table is the algebra $B_{n}=(\{1,...,n\},*)$ where ...
8
votes
0answers
261 views

Model for “$\kappa$ limit cardinal iff $2^\kappa$ limit cardinal”

Is there a model of ${\sf (ZFC)}$ such that in the model we have that $\kappa$ is a limit cardinal if and only if $2^\kappa$ is a limit cardinal?
10
votes
1answer
316 views

'stationary' almost disjoint families

Consider almost disjoint families on regular $\kappa > \omega$ consisting only of stationary sets. My question: Is there consistently an upper bound $<2^\kappa$ on the size of such a '...
1
vote
0answers
131 views

How many critical points can you have below a Fibonacci term in an algebra of elementary embeddings?

In this question, I asked about how slowly the Fibonacci terms in a algebra of elementary embeddings could grow, but I did not inquire if there was a limit to how quickly the Fibonacci terms could ...
4
votes
1answer
271 views

Is each cosmic space cometrizable?

A regular topological space $X$ is called $\bullet$ cosmic if $X$ is a continuous image of a separable metrizable space; $\bullet$ cometrizable if $X$ admits a weaker metrizable topology such that ...
2
votes
0answers
164 views

3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators

During my studies, I came across several different Stone spaces, e.g.: (i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators; ...
8
votes
1answer
213 views

A strictly decreasing function between uncountable subsets of the reals

By a standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G_\delta$-subsets of the real line) it is possible to prove the following ...
11
votes
1answer
765 views

ZF(C) and category theory

Is there an axiomatisation of some kind of category theory and a definition of sets in this framework such that the axioms of ZF resp. ZFC are theorems?
4
votes
1answer
375 views

(non) separability of the power set

Suppose that $(X,2^X)$ is equipped with a non-atomic probability measure $\mu$ (the existence of such spaces is consistent with ZFC). This induces the $L_1$ pseudometric $\Delta$ on $2^X$, via $\...
9
votes
2answers
881 views

Terminology about trees

In set theory, a tree is usually defined as a partial order such that the set of elements below any given one is well-ordered. I am interested in the class of partial orders $P$ such that for every $...
4
votes
1answer
164 views

Are there any I1 embeddings with interweaving critical sequences?

Can there exist non-trivial elementary embeddings $j,k:V_{\lambda+1}\rightarrow V_{\lambda+1}$ along with a strictly increasing function $r:\omega\rightarrow\omega$ such that $j^{r(2n)}(\mathrm{crit}(...
5
votes
1answer
236 views

Variant of Sierpiński's result on non-atomic measures

Sierpiński's theorem states that nonatomic probability measures take a continuum of values. What if I assume that $\mu$ is a countably additive probability measure on $(X,2^X)$ and further that $\mu(\{...
8
votes
0answers
209 views

Topological applications of $\mathfrak{p}=\mathfrak{t}$

I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality. Searching in ...
0
votes
1answer
375 views

Smallest $\beta$ such that it is provable that $2^{\aleph_\beta} > 2^{\aleph_0}$

What is the smallest cardinal $\beta$ such that it is provable in ${\sf (ZFC)}$ that $2^{\aleph_\beta} > 2^{\aleph_0}$?
1
vote
0answers
43 views

A linear ordering on the quotient algebras of elementary embeddings?

We say that a finite self-distributive algebra $(A,*)$ is linear if there is some $1\in A$ where $a*1=1,1*a=a$ for all $a\in A$ and where if $\preceq$ is the relation where $x\preceq y$ if and only if ...
2
votes
0answers
74 views

A Baire space with meager projections

Question. Is there a Baire subspace $X$ of a Tychonoff power $M^\kappa$ of some separable metrizable space $M$ such that for any countable subset $A\subset \kappa$ the projection $$X_A=\{x{\...
2
votes
1answer
730 views

A new generalisation of dimension? part 2

I worked this theory : A new generalization of the dimension? I have a theorem about dimensions which is more general and simple than for matroids. Definition 1: A structure $S$, is a pair $(X, \...
3
votes
0answers
62 views

What is the computational complexity of equivalence up to a critical point in the one generator free self-distributive algebra?

Suppose that there exists a rank-into-rank cardinal. Then let $R$ be the relation on the set of terms in the language of self-distributive algebras (or LD-monoids) on one generator where we set $R(s,t)...
4
votes
1answer
203 views

Does measurability of cardinal $\kappa$ imply measurability of $2^\kappa$?

A cardinal $\kappa$ is real-valued measurable if there is a $\kappa$-additive probability measure on $2^\kappa$ which vanishes on singletons. The existence of measurable $\kappa$ is independent of ZFC....
3
votes
0answers
74 views

Arriving at the critical points in an algebra of elementary embeddings in a unique way

Let $\mathcal{E}_{\lambda}$ denote the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. Then $\mathcal{E}_{\lambda}$ can be endowed with a self-distributive operation $*$ defined ...
9
votes
1answer
202 views

What is known about topological groups of countable spread in ZFC?

A topological space has countable spread if every discrete subspace is at most countable. By Theorem 8.10 in Todorcevic's book "Partition Problems in Topology", PFA implies that each regular space $X$...
7
votes
1answer
292 views

Is a Borel image of a Polish space analytic?

A topological space $X$ is called analytic if it is a continuous image of a Polish space, i.e., the image of a Polish space $P$ under a continuous surjective map $f:P\to X$. We say that a topological ...
3
votes
0answers
66 views

Can algebras of elementary embeddings be sufficiently described by two element subalgebras?

Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Then $\mathcal{E}_{\lambda}$ can be endowed with a self-distributive operation $*$ where $j*...
1
vote
0answers
183 views

Random variables over large measurable cardinals

This question assumes the existence of a large real-valued measurable cardinal. Let $X$ be an uncountable set and $(X,2^X)$ equipped with a non-atomic probability distribution $P$. Additionally, let $...
7
votes
0answers
126 views

If $j_{1},…,j_{n}:V_{\lambda+1}\rightarrow V_{\lambda+1}$ are elementary embeddings, then does $j_{1}(A)=…=j_{n}(A)=A$ for some linear order $A$?

Suppose that $j_{1},\dots,j_{n}:V_{\lambda+1}\rightarrow V_{\lambda+1}$ are elementary embeddings. Then does there necessarily exist a linear ordering $A$ of $V_{\lambda}$ such that $j_{1}(A)=\dots=j_{...
15
votes
1answer
735 views

Axiom of Choice versus V=L in opposition to large cardinals

Consider the following two observations: The axiom $V=L$ is incompatible with large cardinal axioms that are somehow "too large", like measurable cardinals. The axiom of Choice is incompatible with ...
4
votes
0answers
84 views

Examples of Yang-Baxter monoids

Then we say that an algebra $(X,f,g,\circ,1)$ is a Yang-Baxter monoid if it satisfies the following identities: $(X,\circ,1)$ is a monoid, $f(x,1)=1,f(1,x)=x,g(x,1)=x,g(1,x)=1$ $x\circ y=f(x,y)\circ ...
8
votes
0answers
192 views

Proper classes in Bounded Zermelo set theory

I want to know if there is a standard terminology for this among set theorists working with element-based set theories like ZFC. I will follow the convention that a class in any given set theory is a ...
1
vote
0answers
61 views

Consistency of reflective sequences

Is it consistent that there is a measurable cardinal $κ$, a $κ$-complete normal nonprincipal ultrafilter $U$ on $κ$, and $S∈U$ such that for every $T⊂S$ with $T∈U$ and $T$ ordinal definable from $S$ ...
4
votes
0answers
62 views

Permutative Yang-Baxter monoids

Suppose that $f,g:X^{2}\rightarrow X,T:X^{2}\rightarrow X^{2}$ are mappings such that $T(x,y)=(f(x,y),g(x,y))$. An element $1\in X$ is said to be an identity if $T(1,x)=(x,1),T(x,1)=(1,x)$. The ...
3
votes
0answers
131 views

final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?

In the paper Coalgebraic Games and Strategies F. Honsell, M. Lenisa, and R. Redamalla use the functor $F_A$(X) = ${\mathscr{P}_{<κ}}$(A×X) to define games coalgebraically. This is a functor from ...
0
votes
0answers
287 views

What is the consistency strength of this kind of iterating Berkeley cardinals?

[EDIT] After suggestion from Monroe Eskew, and after having an e-mail correspondence with Prof. Joan Bagaria, I'll re-present the older question as the second in a series of 4 questions. I've had an e-...
0
votes
1answer
178 views

Why the restrictions in the definition of Berkeley cardinals?

A cardinal κ is a Berkeley cardinal, if for any transitive set $M$ with $κ∈M$ and any ordinal $α<κ$ there is an elementary embedding $j : M → M$ with $\alpha<\text{crit}(j)<\kappa$. My ...
3
votes
0answers
112 views

Does the Hurwitz action of the braid group on rank-into-rank embeddings tend to increase the critical points?

An algebraic structure $(X,*)$ is said to be self-distributive if it satisfies the identity $x*(y*z)=(x*y)*(x*z)$. Suppose that $X$ is a self-distributive algebra. Then the positive braid monoid $B_{...