# 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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### Systems of elementary embeddings

Recently I've been thinking about elementary embeddings, partition cardinals, etc. as part of my never-ending quest for understanding of consistency strength :p I came up with this idea, called I* ...
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### Are dark numbers known? [closed]

The limit in analysis is approximated better and better by a sequence or series. Contrary to set theory. Bertrand Russell wrote: "But nowadays the limit is defined quite differently, and the ...
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### Definable closure in class-sized expansions of o-minimal groups

I am working in NBG set theory with limitation of size (i.e. the class of all sets is in bijection with the class of ordinals). Let $\mathbf{G}$ be a class-sized o-minimal expansion of an ordered ...
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### Is there a model of each of the following kinds of theories in the first transitive model of ZFC?

The question of whether the minimal transitive model $M$ of $\sf ZFC$ have a model of each consistent extension $\sf T$ of $\sf ZF$, is answered to the negative, basically because $M$ is countable and ...
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### Does every consistent extension of ZF have a model in the minimal transitive model of ZFC?

Suppose there exists a transitive model of $\sf ZFC$. Is it the case that every consistent theory that extends $\sf ZF$ must have a model that is an element of the minimal transitive model of $\sf ZFC$...
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### Are externally pointwise definable models of ZFC subject to the same limitations of the internally pointwise definable ones?

By pointwise definable models, it's meant that every element of those models is definable after a formula in a parameter free manner, but that defining formula is in the language of that model, i.e., ...
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### Proper class of inaccessibles and 1-inaccessible cardinals

Hello I am starting to learn set theory and I am having some difficulty with these notions. Let's say we have a 1-inaccessible cardinal which means it is stronger than the assertion that there is a ...
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### Which model is the minimal pointwise definable model of $\sf ZFC$?

Is the minimal transitive model of $\sf ZFC$ pointwise definable? If not, then what is the minimal pointwise definable model of $\sf ZFC$? Can we define that using Hamkins result for existence of ...
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