# 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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### Skolem's paradox disproof?

Skolem's paradox, derived from the downward Lowenheim-Skolem theorem (dL-S), is that there exists a countable model $M$ of set theory, specifically Zermelo-Fraenkel set theory with the Axiom of Choice ...
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### Bounded subsets of uncoutable totally ordered set

As my username might possibly suggest, set theory and logic is not really an area of mathematics I know much about. But, there is this statement and apparent proof I was able to come up with, both of ...
1answer
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### How do you simplify $\cup\{A, \cup A\}$?

I want to simplify $\cup\{A, \cup A\},$ and also $\cup\cup\{A, \cup A\},$ so forth. I thought $\cup\{A, \cup A\}$ would not be simplified more. To say this in plain english, this union is a set that ...
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### Can we say that any Dedekind-finite family of sets has a choice function without AC?

Let $(S_i)_{i \in I}$ be a family of non-empty sets where $I$ is a Dedekind-finite set. Can we say without the Axiom of Choice that this family has a choice function? I know that for any finite family ...
2answers
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### If $1+ \alpha = \alpha$, $\alpha$ is an infinite ordinal

I need to prove the following statement: If $1+ \alpha = \alpha$, $\alpha$ is an infinite ordinal. I am trying to use Bernstein's Theorem(CBS) to show if $1+ \alpha \leq \alpha$, i.e., there is an ...
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### Is there a name for the family of nested sets generated by $A_{n+1} = \{A_n, f(n+1)\}$ recursion?

In my research, I consider a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and a family of nested sets defined recursively by: $A_{n+1} = \{A_n, f(n+1)\}$ for $n > 1$, and $A_1 = \{ f(1)\}$ as ...
2answers
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### With regards to Zermelo-Fraenkel Set Theory, is the $0$ and $\emptyset$ in $0 =\{ \}= \emptyset$ equal?

I have been studying philosophy for some time now, although my knowledge with respect to mathematics is amateur at best. I recently starting learning set theory on my own, but have ran into an issue (...
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### Some cardinals/ordinals Arithmetic

I am having trouble grasping ideas for the following three problems (and I am unsure whether the third condition even holds). For arbitrary ordinals $\alpha$, $\beta$, (Edit: on the LHS the operations ...
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### How to prove $|B^A| = |B|^{|A|}$ in sets [closed]

If we show the size of set A like this $|A|$ and $A$ and $B$ are infinite sets how can I prove this: $|B^A| = |B|^{|A|}$
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