# Questions tagged [set-theory]

This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.

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### Unions of basis sets in the topology induced by the base

The topology generated by a basis is defined as $\tau=\{U\subset X|\forall x\in U, \exists \beta\subset B, x\in \beta\}$. It is also true that all sets in the topology are unions of basis sets. I have ...
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### Iterated function,Set theory and Collatz conjecture [on hold]

\begin{matrix} \hline & odd& & &even& &\\ \hline 1&2&3&4&5&6\\ \hline n\equiv 0\left ( mod3 \right ) & n\equiv 1\left ( mod3 \right ) & n\equiv 2\...
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### Why is the class of all sets denoted $V$?

In NBG set theory, why is the class of all sets denoted $V$? $S$ seems to me to be the natural designation.
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### Infinite Time Turing Machine and Hypertask

Good Day, I would like to ask this question. Infinite Time Turing Machine ordinals are countable ordinals. Also I have read that ITTM either halts or repeats itselve after countably many steps. Does ...
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### Proving $\|x=y\|\cdot \|\phi(x)\|\le\|\phi(y)\|$ in Boolean valued models

This question relates to the Boolean algebra approach to forcing. Fix a complete Boolean algebra $B$. I'm writing $\|\sigma\|$ for the Boolean value of $\sigma$, where $\sigma$ is a sentence of the ...
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### When is the maximal element guaranteed by Zorn's Lemma unique? [closed]

Zorn's Lemma states that any poset with the property that every chain has an upper has at least one maximal element. Are there necessary or sufficient conditions on the poset for the maximal element ...
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### Some question about the statement of Zorn Lemma

Some textbooks describe the Zorn Lemma as: Every nonempty ordered set S has the maximal element if every totally ordered subset of S has an upper bound in S. Some other books replace the upper bound ...
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### Proof of every measurable cardinal carries a normal measure

I'm reading the proof of Theorem 10.20 in Set Theory by Jech and I don't understand the last argument. The theorem says every measurable cardinal carries a normal measure. The proof goes: Let $U$ be ...