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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.

5
votes
1answer
242 views

Function on the set of limit countable ordinals

Let $\Lambda$ be the set of all countable limit ordinals. Does there exist an injective function $f:\Lambda\to\omega_1$ with the properties: $\forall \lambda\in\Lambda:~f(\lambda)<\lambda$ $\...
2
votes
1answer
72 views

Non-existence of countable base of arbitrary ultrafilter

$\scr{B}$ is the base of a nonprincipal ultrafilter $\scr{U}$ on $\omega$ if 1. $\forall U,V\in\mathscr{B}~\exists T\in\mathscr{B}:~T\subset U\cap V$, 2. $\forall X\in\mathscr{U}~\exists U\in\mathscr{...
0
votes
2answers
83 views

Existence of $\alpha$-sequence of infinitesimal sequences with $\alpha>\omega_1$

We are in ZFC & CH. Given family $Y=\{y_\alpha\}_{\alpha<\omega_1}$ of infinitesimal $\omega$-sequences (i.e. $\lim_{n\to\infty}y_{\alpha n}=0$) of rational numbers with the property: $\forall\...
-2
votes
0answers
111 views

Cardinals from ordinals in ZFC [on hold]

Did someone define the cardinal numbers as the equivalence classes from the relation of bijection on the class On of all ordinal numbers ? Gérard Lang
0
votes
0answers
238 views

Some questions regarding “Elementary Embeddings and Correctness”

Consider the following, from Eskew's (and Friedman's, presumably, though Dr. Eskew's name is under the title) page-long abstract, "Elementary Embeddings and Correctness": Kunen's inconsistency ...
6
votes
0answers
134 views

Complexity of the set of closed subsets of an analytic set

Let $X$ be a compact Polish space and $K(X)$ the hyperspace of closed subspaces of $X$ with the Vietoris/Hausdorff metric topology. Question: If $A$ is an analytic subset of $X$, what is the ...
7
votes
0answers
126 views

Making the precaliber number bigger than all Knaster numbers

Write $\mathfrak m_k$ for the Martin's axiom number for $k$-Knaster, i.e., for the smallest size of a family of dense subsets of some $k$-Knaster poset for which there is no generic filter. (A poset ...
14
votes
1answer
540 views

Do we know the consistency strength of the Singular Cardinal Hypothesis failing on an uncountable cofinality?

Suppose that $\kappa$ is a strong limit cardinal. The singular cardinal hypothesis states $2^\kappa=\kappa^+$. We know that the failure of SCH requires large cardinals, and in fact is equiconsistent ...
3
votes
1answer
135 views

K-analytic spaces whose any compact subset is countable

A regular topological space $X$ is called $\bullet$ analytic if $X$ is a continuous image of a Polish space; $\bullet$ $K$-analytic if $X$ is the image of a Polish space $P$ under an upper ...
12
votes
1answer
383 views

End-extending cardinals

Let us say a cardinal $\kappa$ end-extending if there is a function $F : V_\kappa^{<\omega} \to V_\kappa$ such that: (a) If $M \subseteq V_\kappa$ is closed under $F$, then $M \prec V_\kappa$. (b) ...
2
votes
2answers
454 views

Who first discovered the concept corresponding to the symbol of class comprehension?

Who first discovered the concept corresponding to the symbol of class comprehension {x/𝛗}used today in set theory ?
3
votes
1answer
124 views

Radin forcing preserving large cardinals

I'm wondering if there are any known result for the maximum large cardinal strength which can be preserved by Radin forcing? For instance, with any large cardinal hypothesis in the ground model, can ...
13
votes
2answers
599 views

Set-theoretical foundations of Mathematics with only bounded quantifiers

It seems that outside of researchers in Mathematical Logic, mathematicians use almost exclusively bounded quantifiers instead of unbounded quantifiers. In fact, I haven't observed any other practice ...
2
votes
1answer
176 views

Proving that family of sets has non-empty intersection

Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already: $S$ is set of measurable ...
5
votes
1answer
228 views

Number of ultrafilters in an extender

Assume GCH. Suppose $j: V\to M$ is an elementary embedding such that $\mathrm{crit}(j)=\kappa$, ${}^\kappa M \subset M$, $M\supset V_{\kappa+2}$. We can assume $j$ is defined from some extender of ...
5
votes
1answer
203 views

Radin generics from iterated ultrapowers

Let $u$ be a measure sequence derived from some embedding $j:V\rightarrow M$. I heard that from iterating $j$ is possible to define a generic filter for some Radin forcing $\mathbb{R}_w$. Specifically,...
3
votes
0answers
167 views

A consistent way to “decide” independent statements [closed]

Let us fix some mathematical theory T=T(0), such as ZFC. The aim is to develop algorithm A, which takes any statement S independent of T(0) as an input, and outputs axiom a(S) such that T(1)=T(0)+a(S) ...
2
votes
0answers
50 views

Cardinal characteristics of the ideal of $\sigma$-continuity of the Pawlikowski function

A function $f:X\to Y$ between topological spaces is called $\sigma$-continuous if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\...
7
votes
1answer
412 views

Cardinal characteristics of amorphous sets

In a universe where the continuum hypothesis ($CH$) fails we can ask about combinatorial cardinal characteristics of the continuum, but in a universe where $CH$ is true no such cardinals exist so this ...
3
votes
0answers
208 views

Is there a standard meaning for this notation for ordinals in set theory?

In the paper ``Berkeley Cardinals and the Structure of $L(V_{\delta+1})$", by Raffaella Cutolo, a use is made of the notation $\alpha < < \beta$ for a pair of ordinals $\alpha, \beta$, without ...
1
vote
0answers
86 views

Two small uncountable cardinals related to Q-sets

A subset $A$ of the real line is called a Q-set if any subset of of $A$ is of type $F_\sigma$ in $A$. Let $\mathfrak q_0$ be the smallest cardinality of a subset $X\subset\mathbb R$ which is not a Q-...
2
votes
1answer
73 views

The trace of the filter on a big subset

Let $\scr{F}$ be free filter ($\cap\scr{F}=\emptyset$) on a countable set $X$ and $B\in\scr{F}$. We define the trace of $\scr{F}$ on $B$ as follows $\mathscr{F}_B=\{Y\cap B:~Y\in\scr{F}\}$. $\scr{F}$ ...
5
votes
2answers
213 views

Ordered union of Borel sets

Let $\mathfrak{A}$ be an uncountable collection of Borel sets in $\mathbb{R}^d$ such that for any $A,B\in\mathfrak{A}$, either $A\subset B$ or $A\supset B$. Then is it necessarily true that the union $...
-2
votes
0answers
105 views

Can every first order theory have a finitely axiomatizable conservative extension in this sense?

Note: This is an edit of the previous question. By $T_2$ conservatively extends $T_1$ if and only if: There exists a function $F$ such that $T_2$ extends $T_1$ through $F$; and for every function $G$...
0
votes
0answers
58 views

Existance of bijective function which maps tensor product of subsets of a selective ultrafilter into the ultrafilter

In the answer on this question Andreas Blass had shown that for any selective ultrafilter $\scr{U}$ on $\omega$ and for any free subfilter $\scr{F}\subset{U}$ doesn't exist bijection $\varphi:\omega^2\...
21
votes
1answer
701 views

Does “every” first-order theory have a finitely axiomatizable conservative extension?

I originally asked this question on math.stackexchange.com here. There's a famous theorem (due to Montague) that states that if $\sf ZFC$ is consistent then it cannot be finitely axiomatized. ...
7
votes
0answers
130 views

Asymptotically discrete ultrafilters

Definition 1. A ultrafilter $\mathcal U$ on $\omega$ is called discrete (resp. nowhere dense) if for any injective map $f:\omega\to \mathbb R$ there is a set $U\in\mathcal U$ whose image $f(U)$ is a ...
10
votes
0answers
382 views

Sunflower / $\Delta$-system lemma in a more general poset?

The sunflower lemma (or $\Delta$-system lemma) may be viewed as a statement about the poset $P_\omega(\omega_1)$, and the generalized sunflower lemma may be viewed as a statement about the poset $P_\...
5
votes
1answer
182 views

Relation between ultrafilters ${\scr U}$ and ${\scr U} \otimes {\scr U}$ [closed]

If ${\scr U}$ and ${\scr V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the tensor product ${\scr U}\otimes{\scr V}$ is the following ultrafilter on $A\times B$: $$\big\{X\...
-3
votes
0answers
334 views

Can we get rid of the primitive symbol $V$ in Ackermann's set theory without increment in consistency strength?

EDIT: Ackermann had presented his theory with a new primitive added to the language of set theory that is the "set-hood" primitive one place predicate symbol $\mathcal M$ or in common equivalent ...
2
votes
1answer
73 views

Dense subfilter of selective ultrafilter

Given selective ultrafilter $\mathcal{U}$ on $\omega$ and dense filter $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$, where $\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$ if the limit exists. Let $\...
5
votes
1answer
236 views

Models of $\mathsf{ZFC}$ with neither $P$- nor $Q$-points

A $P$-point is an ultrafilter $\scr U$on $\omega$ such that for every function $f:\omega\to\omega$ there is $x\in {\scr U}$ such that the restriction $f|_x$ is either constant, or finite-to-one. A $Q$...
6
votes
1answer
219 views

Finite covers of Boolean algebras by their subalgebras

It is a student exercise that no group can be represented as a set-theoretic union of its two proper subgroups. The same also can be shown for Boolean algebras. On the other hand, it's not hard to ...
-1
votes
0answers
56 views

Largest subset of the powerset of a countable set in which no set includes another [duplicate]

Let S be a set that has countably-infinitely many members. Let a subset of $\mathcal{P}(S)$ (the power-set of S) have the Sperner-family-property iff no two of its members are such that one of them is ...
0
votes
0answers
121 views

Subsets of the unit interval [migrated]

I have been working on a problem and I need to answer the following question: Is there a family $\{F_\alpha: \alpha \in \omega_1\}$ of subsets of the interval $]0,1[$ such that: (a) $F_\alpha=\{x_1^\...
2
votes
2answers
241 views

(Types of) induction on infinite chains

This question may be trivial, or overly optimistic. I do not know (but I guess the latter...). I am a group theorist by trade, and the set-up I describe cropped up in something I want to prove. So ...
5
votes
0answers
191 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 ...
5
votes
1answer
163 views

The property of the dense subfilter of a selective ultrafilter

Let us define the density of subset $A\subset\omega$ : $$\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$$ if the limit exists. Let $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$. $\mathcal{F_1}$ is the ...
1
vote
1answer
98 views

Some kind of idempotence of dense filter

In discussion of following questions question1, question2, question3 became clear (see definitions in question3 ) that for the Frechet filter $\mathcal{N}$ we have $\mathcal{N}\nsim\mathcal{N}\otimes\...
1
vote
1answer
110 views

Semi-rigid boolean algebras

A boolean algebra is rigid if it has no nontrivial automorphisms. Call it semi-rigid if none of its nontrivial automorphisms has any fixed points other than 0 and 1.* The four-element algebra $\{0, b, ...
4
votes
0answers
95 views

Is Ackermann's set theory minus class comprehension equal to ZF?

Ackermann in 1956 proposed an axiomatic set theory. Reinhard proved that Ackermann's set theory equals ZF It's clear that Zermelo set theory can be interpreted in Ackermann's set theory minus class ...
5
votes
1answer
222 views

Amorphous proper classes in MK

Working in $ZFC$ every cardinal is either finite or in bijection with a proper subset of itself (Dedekind infinite). Without Choice it is consistent that there are infinite sets which can't be ...
40
votes
0answers
603 views

How many algebraic closures can a field have?

Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is ...
15
votes
1answer
282 views

Is Global Choice conservative over Zermelo with Choice?

To be explicit, by Zermelo set theory with Choice, ZC, I mean the theory with the same language and axioms as ZFC except not Foundation (also called Regularity) and with the axiom scheme of Separation ...
0
votes
1answer
108 views

Finitely additive measure on Cartesian square of countable set

Let $\mu$ be a probability measure on $(\omega, 2^\omega, \mu)$ measure space which is finitely additive and $\mu(A)=0$ for finite sets. We can define as usual $\mu^2$ on semiring $\mathcal{G}=\{A\...
9
votes
2answers
477 views

Examples of transfinite towers

I am looking for examples of constructions for transfinite towers $(X_{\alpha})_{\alpha}$ generated by structures $X$ where the problem of determining whether the tower $(X_{\alpha})_{\alpha}$ stops ...
1
vote
1answer
97 views

Maximal elements in the Rudin-Keisler ordering

Let $\text{NPU}(\omega)$ be the set of non-principal [ultafilters][1] on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by $${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\...
10
votes
2answers
341 views

On the absoluteness of higher Borel sets?

Consider the higher Cantor space $2^\kappa$ with the ${<}\kappa$-box topology ($\kappa$ at least inaccessible). This canonically defines the notion of higher Borel sets. A higher Borel code $\...
1
vote
0answers
69 views

Why does $p_{n}(i,1)=1$ so often where the polynomials $p_{n}$ are obtained from the classical Laver tables

So I was doing some computer calculations with the classical Laver tables and I found polynomials $p_{n}(x,y)$ such that $p_{n}(i,1)=1$ for many $n$. The $n$-th classical Laver table is the unique ...
0
votes
0answers
115 views

Infinite products of complex numbers or matrices arising from rank-into-rank embeddings

I wonder what kinds of closed form infinite products of matrices, elements of Banach algebras, and complex numbers arise from the rank-into-rank embeddings. Suppose that $\lambda$ is a cardinal and $...