# 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.

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### 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 ...
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### 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 ...
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### 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 ...
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### What is the consistency strength of this kind of reflection principle?

If $\psi$ is a predicate that is definable in $FOL(\in,=)$ by a formula from parameters in $V$, then if $\psi$ hold of the class $\small ORD$ of all ordinals in $V$, then the class of all cardinals in ...
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### Embeddability into $\beta\omega$ and $\omega^*$

It is well known that under CH every totally-disconnected compact F-space of weight at most $\omega_1$ can be embedded into the remainder $\omega^*=\beta\omega\setminus\omega$ of the Cech-Stone ...
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### Non-discrete $T_2$-space $(X,\tau)$ with $2^{|X|}$ retracts

If $(X,\tau)$ is a topological space, we call $A\subseteq X$ a retract if there is a continous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$ (we assume $A$ to be endowed with the subspace ...
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Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ... 3answers 953 views ### How many sigma algebras exist on$\mathbb{R}$? On the one hand, there are at least$2^\mathfrak{c}$sigma algebras on$\mathbb{R}$: one can take any subset$A$of$\mathbb{R}$and consider a sigma algebra$\{\emptyset, A, \bar A, \mathbb{R}\}$On ... 1answer 146 views ### What is the strength of this strict constructible iterative hierarchy? Begin with the empty set then construct the set of the empty set, then construct the set of all subsets of the latter set, then at each level of construction construct the next level as the set all ... 1answer 200 views ### Is Replacement motivated by ranked iterative conception of sets? When one reads the Wikipedia article on the Von Neumann Universe, one gets the impression that the idea of "the cumulative hierarchy" serves as a motivation for$ZFC$. I don't see really how this is ... 1answer 517 views ### Is there a pair of non-isomorphic structures each of which is isomorphic to an ultrapower of the other? Does there exist a pair of non-isomorphic structures$\mathfrak{A}$and$\mathfrak{B}$as well as sets$I$and$J$and ultrafilters$\mathcal{U}$on$I$and$\mathcal{F}$on$J$such that$\mathfrak{A}...
An ideal $I$ on $P(\kappa)$ is said to be $\kappa^+$-saturated if there is no sequence $\langle S_\alpha \mid \alpha<\kappa^+\rangle$ of $I^+$ sets such that \$\alpha<\beta<\kappa^+\implies S_\...