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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.

2
votes
0answers
63 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 ...
-1
votes
1answer
224 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
33 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
67 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{\...
3
votes
1answer
515 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, \...
2
votes
0answers
48 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)...
3
votes
1answer
167 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....
2
votes
0answers
63 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 ...
4
votes
0answers
87 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
191 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
64 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
159 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 $...
6
votes
0answers
115 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
649 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
68 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
177 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
56 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
51 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
124 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 ...
-1
votes
1answer
221 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
157 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
103 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_{...
19
votes
1answer
779 views

Fubini without CH

In Real and Complex Analysis, Rudin gives an example (due to Sierpinski) of a function $f:[0,1]^2\to[0,1]$ separately Lebesgue-measurable in each argument, such that $$ \int_0^1 dx\int_0^1f(x,y)\,dy \...
3
votes
1answer
129 views

Induced minors of $\{0,1\}^\omega$

Let $G=(V,E)$ be a simple, undirected graph. Suppose that ${\cal S}$ is a collection of non-empty, connected, and pairwise disjoint subsets of $V$. Let $G({\cal S})$ be the graph with vertex set ${\...
0
votes
1answer
141 views

Axiom of Regularity (set theory) - trying to understand it [closed]

I'm doing my first steps in set theory and have a question about the Axiom of Regularity. AoR states: $\forall x (x\neq \emptyset \rightarrow \exists (y \in x) x \cap y = \emptyset)$. What causes my ...
-3
votes
1answer
187 views

Can there be elementary embedding between a universe and a universe inside it?

[EDIT] the prior question was trivially false, however the intention is to arrange a possible world of such universes, in other words the question is about if it is possible to have a proper class $\...
0
votes
1answer
76 views

Strength of BTEE

What is the consistency strength of Basic Theory of Elementary Embeddings (BTEE) from The spectrum of elementrary embeddings j : V → V by Paul Corazza? BTEE uses the language of $(V,∈,j)$ and asserts:...
0
votes
1answer
71 views

Finding a good transversal basis

A hypergraph $H=(V,E)$ consists of an non-empty set $V$ and a collection $E\subseteq {\cal P}(V)\setminus \{\emptyset\}$ of non-empty subsets of $V$. A transversal of $H$ is a set $T\subseteq V$ such ...
0
votes
0answers
87 views

Is there a clear inconsistency with intense reflection of top properties of the universe?

Let $V$ be the class of all sets, where sets are defined like in $MK$ as elements of classes. Properties of $V$ whose negations are unbounded (by element-hood & subset-hood) in $V$ would be ...
0
votes
0answers
260 views

What is the consistency status of this theory?

Let $K_2^+(W)$ be the following theory in the language $L(\in,W)$ with the constant symbol $W$. Extensionality: $\forall x (x \in a \leftrightarrow x \in b) \to a=b$ $\mathcal{Define:} \ set(x) \iff ...
8
votes
0answers
152 views

α-Mahlo vs weakly compact cardinals

Question: What is the consistency strength of existence of a $(κ^{++})^L$-Mahlo cardinal $κ$? I am particularly interested in how the strength compares to weakly compact cardinals (and other levels ...
16
votes
1answer
510 views

How is this fixed point theorem related to the axiom of choice?

I'm hoping the answer to this is well-known. Let $X$ be an ordered set (i.e. poset). An inflationary operator $f$ on $X$ is a function $f: X \to X$, not necessarily order-preserving, such that $f(x) ...
2
votes
1answer
306 views

Can Ackermann set theory find a natural interpretation in light\heavy class dichotomy?

I want to coin the notion of heaviness of a class as a function from classes to ordinals such that $$heaviness(x)=|TC(x)|$$, i.e. the heaviness of a class is the cardinality of its transitive closure. ...
11
votes
0answers
236 views

Consistency strength of $j:L_δ→L_δ$ for some δ

What is the consistency strength of existence of a nontrivial elementary embedding $j:L_δ→L_δ$ for some ordinal $δ$? The consistency strength is strictly between totally ineffable and $ω$-Erdős ...
0
votes
0answers
62 views

Is Reflective Set Theory stronger than $\small {\mathsf{ARC}}$?

By $\mathsf{RfST}$ its meant Reflective Set Theory exposited in this posting I'll pose two specific questions here: Is $\mathsf{ARC}$ class theory a proper sub-theory of $\mathsf{RfST}$? Is $...
1
vote
0answers
236 views

Does this axiomatic system satisfy requirements for founding mathematics?

In this article, the author, F.A.Muller, suggests criteria for a founding theory of mathematics (pp:14-16). The author proposes $ARC$ Class Theory to embody these requirements. The motivation is ...
3
votes
0answers
163 views

Arithmetic sums of Marczewski null sets

First, recall the Marczewski ideal, called $s^0$: a set $A$ of reals is in $s^0$ iff for every perfect set $P$ there is some perfect $P' \subset P$ such that $P' \cap A = \emptyset$. Secondly, by way ...
5
votes
1answer
182 views

Ordinal Exponentiation Levy Hierarchy

It is a standard exercise (see Jech's "Set Theory" Exercise 13.8) to prove that ordinal addition and multiplication are $\Delta_1$ expressible functions. The proof for addition comes from noting that ...
10
votes
0answers
559 views

Toposophy vs Set theoretical multiverse philosophy

Johnstones classic topos theory book talks at some length in its introduction about how category theory/topos theory suggest that we view the 'universe' in which mathematics takes place as consisting ...
1
vote
1answer
136 views

On a combinatorial set covering property

Let $\kappa < \lambda < \mu$ be infinite cardinals. Is there a collection ${\cal U}\subseteq {\cal P}(\mu)$ of subsets of $\mu$ with the following properties? for all $U\in {\cal U}$ we have $|...
2
votes
1answer
168 views

Another question concerning p and t

I refer to an article concerning p and t : edited Sep 14 '17 at 2:48 / Bjørn Kjos-Hanssen answered Sep 13 '17 at 21:50 / Mark Fischler I already asked a question December 14th 2018 and I received ...
0
votes
0answers
101 views

Can reflection in $V$ and reflection out of $V$ principles be used in the same theory?

I've noticed that most of the work in Ackermann set theory is primarily concerned with constructing sets in $V$, the rest of the classes are just excess material, carrying no comprehension over them. ...
4
votes
0answers
106 views

Can this reflective class theory interpret ZFC?

Reflective Set Theory $\mathsf{RfST}$ is formulated in first order predicate logic with extra-logical primitives of equality $``="$, membership $``\in"$, and a single primitive constant symbol $V$ ...
7
votes
1answer
182 views

Effective set= ordinal definable set

I just today realized that the concept of ordinal definability is defined in a different way by vopenka-Balcar-Hajek ``The notion of effective sets and a new proof of the consistency of the axiom of ...
-1
votes
0answers
62 views

Why is $\omega^{\omega}$ countable? [migrated]

I'm confused as to why $|\omega^{\omega}| \neq \aleph_0^{\aleph_0}$. Since \begin{align} \omega^{\omega} = \left\lbrace \sum_{i < \omega}^{1} (\omega^i \cdot n_i) + n_0 : n_i,n_0 \in \mathbb{N}_0 \...
0
votes
0answers
56 views

What is the limit to iterating class comprehension, reflection and limitation of size?

In posting about a reflection principle coupled with a limitation of size axiom over Ackermann set theory, the answer is that the theory is blown up to a Mahlo cardinal. I'm here just wondering if ...
8
votes
0answers
168 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 $$ \...
1
vote
0answers
546 views

A new generalization of the dimension?

During my research, I came a cross on these notions : Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
3
votes
1answer
154 views

What is the strength of adding limitation of size and a simple version of reflection to Ackermann set theory?

The following theory is formulated in first order predicate logic with extra-logical primitives of equality $``="$, membership $``\in"$, and a single primitive constant symbol $V$ denoting the class ...
6
votes
0answers
254 views

measure of generic reals in forcing extensions

It is well-known that if $V[G]$ is a generic extension by adding a Cohen real, then the set $\{r \in V[G]: r$ is Cohen generic over $V\}$ has measure zero. On the other hand, if $V[G]$ is a generic ...