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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.

9
votes
1answer
267 views

A question on the ultrafilter number

Let $\mathfrak{u}$ denote the ultrafilter number, which is defined to be the minimum cardinality of a subset of $\mathcal{P}(\mathbb N)$ which is a base for a nonprincipal ultrafilter on $\mathbb{N}$. ...
2
votes
0answers
86 views

Covering numbers - looking for a more combinatorial proof

For cardinals $\mu$, $\kappa$, $\theta$, and $\sigma$, the covering number $cov(\mu,\kappa,\theta,\sigma)$ is defined to be the minimum cardinality of a set $P\subseteq [\mu]^{<\kappa}$ such that ...
7
votes
1answer
140 views

Non-tensor-representable ultrafilters on $\omega$

If ${\cal U}$ and ${\cal V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the tensor product ${\cal U}\otimes{\cal V}$ is the following ultrafilter on $A\times B$: $$\big\{X\...
7
votes
1answer
217 views

Is there an abstract theory of club sets and stationary sets?

In set theory, there are several distinct notions of club sets, stationary sets, diagonal intersection, regressive function, normal filters, normal ultrafilters, etc. I am wondering if there is an ...
3
votes
1answer
104 views

Minimal cardinality of a filter base of a non-principal uniform ultrafilters

Let $\kappa$ be an infinite cardinal. An ultrafilter ${\cal U}$ on $\kappa$ is said to be uniform if $|R|=\kappa$ for all $R\in{\cal U}$. If ${\cal U}$ is a non-principal ultrafilter on $\kappa$, ...
1
vote
0answers
39 views

Growth rate of the critical points of the Fibonacci terms $t_{n}(x,y)$ vs $t_{n}(1,1)$ in the classical Laver tables

The classical Laver table $A_{n}$ is the unique algebra $(\{1,\dots,2^{n}\},*_{n})$ where $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ and $x*_{n}1=x+1\mod 2^{n}$ for all $x,y,z\in A_{n}$. Define the ...
1
vote
0answers
28 views

Attraction in Laver tables

If $X$ is a self-distributive algebra, then define $x^{[n]}$ for all $n\geq 1$ by letting $x^{[1]}=x$ and $x^{[n+1]}=x*x^{[n]}$. The motivation for this question comes from the following fact about ...
8
votes
1answer
418 views

Axiom of choice and algebraic tensor product

The first part of the question was asked on Math-stackexchange. Let $V$, and $W$ be vector spaces. By the universal property of the tensor product, there is a canonical map from $V^*\otimes W^*$ ...
5
votes
1answer
180 views

Amalgamation via elementary embeddings

Can there exist three transitive models of ZFC with the same ordinals, $M_0,M_1,N$, such that there are elementary embeddings $j_i : M_i \to N$ for $i<2$, but there is no elementary embedding from $...
1
vote
0answers
46 views

Multiple roots in the classical Laver tables

The classical Laver table $A_{n}$ is the unique algebraic structure $$(\{1,\dots,2^{n}\},*_{n})$$ such that $x*_{n}1=x+1\mod 2^{n}$ and $$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$$ for all $x,y,z\in\{1,...
4
votes
0answers
83 views

Universal and strong $Q$-sets

A subset $X\subset \mathbb R$ is called $\bullet$ a $Q$-set if for every subset $A\subset X$ there exists a $\sigma$-compact set $C\subset\mathbb R$ such that $C\cap X=A$; $\bullet$ a strong $Q$-set ...
4
votes
1answer
631 views

Is this lemma equivalent to the axiom of choice?

Given any pre-ordering $\preceq$ of an arbitrary set $X$ is the following lemma: $$\text{There exists an inclusion minimal set }S\text{ satisfying }\{a\preceq b:b\in S\}=X\\\iff \text{ Every chain in ...
1
vote
1answer
72 views

Maximizing “happy” vertices in splitting an infinite graph

This question is motivated by a real life task (which is briefly described after the question.) Let $G=(V,E)$ be an infinite graph. For $v\in V$ let $N(v) = \{x\in V: \{v,x\}\in E\}$. If $S\subseteq ...
4
votes
1answer
83 views

The example of the idempotent filter or subsets family with finite intersections property

From the answer of Andreas Blass and comments of Ali Enayat on my question Selective ultrafilter and bijective mapping it became clear that for any free ultrafilter $\mathcal{U}$ we have $\mathcal{U}\...
1
vote
0answers
39 views

Can we have $\sup\{\alpha\mid(x*x)^{\sharp}(\alpha)>x^{\sharp}(\alpha)\}=\infty$ in an algebra resembling the algebras of elementary embeddings?

A finite algebra $(X,*,1)$ is a reduced Laver-like algebra if it satisfies the identities $x*(y*z)=(x*y)*(x*z)$ and if there is a surjective function $\mathrm{crit}:X\rightarrow n+1$ where $\mathrm{...
1
vote
0answers
39 views

In the classical Laver tables, do we have $o_{n}(1)<o_{n}(2)$ for any $n>8$?

The classical Laver table $A_{n}$ is the unique algebraic structure $(\{1,\dots,2^{n}\},*_{n})$ where $$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$$ and where $$x*_{n}1=x+1\mod 2^{n}$$ for $x,y,z\in\{1,\...
3
votes
1answer
133 views

Nonexistence of a 'product universal' compact Hausdorff pseudotopological space?

I'm largely following the definitions of this paper, but I will replicate the relevant ones here. I'm taking a pseudotopological space to be a set $X$ together with a relation $\rightarrow$ on the ...
2
votes
0answers
34 views

What possible order type can the critical points of these algebras with one generator achieve?

Suppose that $(X,*)$ is an algebra that satisfies the self-distributivity identity $x*(y*z)=(x*y)*(x*z)$. We say that an element $x\in X$ is a left-identity if $x*y=y$ for all $x\in X$. Let $\mathrm{...
9
votes
2answers
340 views

Small uncountable cardinals related to $\sigma$-continuity

A function $f:X\to Y$ is defined to be $\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ ...
5
votes
1answer
747 views

Which branches of mathematics can be done just in terms of morphisms and composition?

Consider the first-order language $L_{\omega\omega}$ of the signature $L:=\{\mathrm{dom}, \mathrm{cod}, \mathrm{comp}\}$, where $\mathrm{dom}$ and $\mathrm{cod}$ are unary function symbols and $\...
7
votes
1answer
260 views

On infinite combinatorics of ultrafilters

Let $\mathcal{U}$ be (selective) ultrafilter and $\omega = \coprod_{i<\omega}S_i$ be partition of $\omega$ with small sets $S_i\notin\mathcal{U}$. All $S_i$ are infinite. Does there exist a system ...
2
votes
0answers
68 views

For each $n$ is it possible to have $\mathrm{crit}(x^{[n]}*y)>\mathrm{crit}(x^{[n-1]}*y)>\dots>\mathrm{crit}(x*y)$?

Suppose that $(X,*,1)$ satisfies the following identities: $x*(y*z)=(x*y)*(x*z),1*x=x,x*1=1$. Define the Fibonacci terms $t_{n}(x,y)$ for $n\geq 1$ by letting $$t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=...
1
vote
0answers
28 views

Vastness of inverse systems of Laver-like algebras

Suppose that $(X,*,1)$ satisfies the identities $x*(y*z)=(x*y)*(x*z),x*1=1,1*x=x$. Then we say that $(X,*,1)$ is a reduced Laver-like algebra if whenever $x_{n}\in X$ for all $n\in\omega$, there is ...
1
vote
0answers
21 views

Can we always extend a finitely generated reduced Laver-like algebra to a vast inverse system of Laver-like algebras?

An $(X,*,1)$ that satisfies the identities $x*(y*z)=(x*y)*(x*z),1*x=x,x*1=1$ is said to be a reduced Laver-like algebra if whenever $x_{n}\in X$ for $n\in\omega$, there is some $N\in\omega$ where $x_{...
16
votes
2answers
363 views

Raising the index of accessibility

In the standard reference books Locally presentable and accessible categories (Adamek-Rosicky, Theorem 2.11) and Accessible categories (Makkai-Pare, $\S$2.3), it is shown that for regular cardinals $\...
-4
votes
2answers
352 views

Is the notion of measurable cardinal definable from the perspective of set-theoretical potentialism?

Consider the definition of measurable cardinal (this definition was found in Neil Barton's paper, "Large cardinals and the iterative conception of set"): Definition 8. A cardinal $\kappa$ is ...
10
votes
1answer
237 views

Selective ultrafilter and bijective mapping

For arbitrary (selective) ultrafilter $\mathcal{F}$ does there exist bijection $\phi:[\omega]^2\to\omega$ with the property: $\forall B\in\mathcal{F} : \phi([B]^2)\in\mathcal{F}$ ?
4
votes
1answer
327 views

Surreal numbers and the Axiom of Choice

In ZFC and its conservative extension NBG, it can be shown that every ordered field embeds into the surreal numbers. How much choice is needed to prove this? Without choice, what is a simple example ...
1
vote
0answers
65 views

Generalization of the linear extension theorem to directed acyclic graphs

Using Zorn's lemma one can prove a generalization of the order extension theorem, that states any acyclic digraph is always contained in another acyclic unilaterally connected digraph on the same ...
6
votes
1answer
217 views

Locally presentable categories, universes, and Vopenka's principle

Some aspects of the theory of locally presentable categories depends on set-theoretic assumptions. Namely, a large cardinal axiom known as Vopenka's principle implies several nice properties of such ...
3
votes
0answers
139 views

Ordering large cardinal axioms around the level of $n$-huge by consistency strength?

So the large cardinal axioms are for the most part considered to be linearly ordered by consistency strength. For the large cardinals between extendibility and rank-into-rank (i.e. the $n$-huge ...
2
votes
0answers
42 views

Calibrating the strength of the quotients of subalgebras of the classical Laver tables

Define an algebraic structure $A_{n}$ by letting $$A_{n}=(\{1,\dots,2^{n}-1,2^{n}\},*_{n})$$ where $*_{n}$ is the unique operation such that $x*_{n}1=x+1\mod 2^{n}$ for $$x\in\{1,\dots,2^{n}-1,2^{n}\}$...
2
votes
2answers
216 views

Are there $2^{\aleph_0}$ pairwise non-isomorphic Boolean algebras on $\omega$?

Is there a collection of $2^{\aleph_0}$ pairwise non-isomorphic countable Boolean algebras? Equivalently, are there $2^{\aleph_0}$ pairwise non-homeomorphic closed subsets in the Cantor space?
5
votes
1answer
403 views

Existence of a strange function

Inspired by A discontinuous construction: Does there exist a function $a \colon [0,1] \to (0,\infty)$ and a family $\{D_x \colon x \in [0,1]\}$ of countable, dense subsets of $[0,1]$ with $\bigcup_{x \...
0
votes
0answers
43 views

Are these conditions sufficient for a self-distributive algebra to occur in the algebras of elementary embeddings?

Suppose that $\mathcal{E}_{\lambda}$ is the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$ and $*$ is the operation on $\mathcal{E}_{\lambda}$ defined by $j*k=\bigcup_{\alpha&...
5
votes
1answer
315 views

Is it consistent that $|[\kappa]^{<\kappa}| > \kappa$?

Let $\kappa>0$ be a cardinal, and let $[\kappa]^{<\kappa}$ denote the collection of subsets of $\kappa$ having cardinality strictly less than $\kappa$. Is it consistent that $$|[\kappa]^{<\...
7
votes
1answer
531 views

Explaining the consistency of PRA and ZF from predicative foundations

Recently I got interested in predicative foundations, mostly because of Laura Crosilla's work and because Agda employs a predicative type theory. From the point of view of a predicative foundation to ...
0
votes
1answer
72 views

Linear intersection number and chromatic number for infinite graphs

Given a hypergraph $H=(V,E)$ we let its intersection graph $I(H)$ be defined by $V(I(H)) = E$ and $E(I(H)) = \{\{e,e'\}: (e\neq e'\in E) \land (e\cap e'\neq \emptyset)\}$. A linear hypergraph is a ...
1
vote
0answers
26 views

Density of different types of critical points in an algebra of elementary embeddings

Suppose that $j,k:V_{\lambda}\rightarrow V_{\lambda}$ are elementary embeddings. Let $\mathrm{crit}_{n}(j,k)$ denote the $n$-th element in $\{\mathrm{crit}(\ell)\mid\ell\in\langle j,k\rangle\}$. ...
1
vote
0answers
29 views

Density of critical points subalgebras of the algebras of elementary embeddings

Let $j:V_{\lambda}\rightarrow V_{\lambda}$ be an elementary embedding. Then $\{\mathrm{crit}(k)\mid k\in\langle j\rangle\}$ has order type $\omega$, so let $\mathrm{crit}_{n}(j)$ denote the $n$-th ...
6
votes
0answers
173 views

Can one use forcing as a step to prove the Keisler-Shelah isomorphism theorem?

Can one use forcing (perhaps at the expense of using larger ultrafilters) to prove the Keisler-Shelah isomorphism theorem? My idea is to use an ultrafilter $U$ on a set $I$ such that if $M=V^{I}/U$, ...
9
votes
0answers
175 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 ...
6
votes
1answer
321 views

Do any finite predictions of Quantum Mechanics depend on the set theoretic axioms used?

I was wondering if any of the finite predictions of Quantum Mechanics depend on what set theoretic axioms are used. We will say that Quantum Mechanics makes a finite prediction about an experiment if,...
3
votes
0answers
105 views

Partial well-ordering of formulas

Given a theory $T$, for arbitrary formulas $φ$ and $ψ$ that provably in $T$ denote an ordinal, set $[φ]_T < [ψ]_T$ iff provably in $T$, the ordinal denoted by $φ$ is less than the ordinal denoted ...
1
vote
0answers
30 views

Is every critically subsimple Laver-like algebra a quotient of a critically simple Laver-like algebra on the same number of generators?

A finite reduced Laver-like algebra is a finite algebra $(X,*,1)$ that satisfies the identities $1*x=x,x*1=1,x*(y*z)=(x*y)*(x*z)$ and where there is a natural number $n$ and a function $\mathrm{crit}:...
13
votes
1answer
309 views

What are some good lower bounds on the consistency of the failure of the PCF conjecture?

Shelah's celebrated theorem states that $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega}<\aleph_{\omega_4}$. But the conjecture is that $\omega_4$ can be provably replaced by $\...
4
votes
0answers
142 views

Symmetry between V and HOD

Is it consistent that the set of ordinal definable real numbers is countable, but for every $y∈\mathrm{OD}∩ℝ$, every true $Σ_2^{V,y}$ statement holds in $\mathrm{HOD}$? Note that $Σ_2^V$ is the best ...
3
votes
0answers
77 views

Why do highly composite rows on the bad Laver tables have longer periods?

For all natural numbers $n$, let $(B_{n},*_{n})$ be the algebraic structure with underlying set $\{1,\dots,n\}$ where $x*_{n}1=x+1\mod n$, $n*_{n}y=y$, and $x*_{n}(y+1)=(x*_{n}y)*_{n}(x+1)$ for $x<...
3
votes
1answer
203 views

Does every directed graph have a directed coloring with $4$ colors?

Every finite directed graph has a majority coloring with $4$ colors. (The notion of majority coloring is defined below.) Question. Can every infinite directed graph be majority-colored with $4$ ...
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 ...