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.

This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. More elementary questions should use the tag instead.

Topics relevant to this tag include cofinality, axioms of ZFC and of close variants (such as ZFA, KP, NBG, MK), 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.

For questions about alternative set theories, use instead the tag .

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Why is the axiom of choice separated from the other axioms?

I don't know much about set theory or foundational mathematics, this question arose just out of curiosity. As far as I know, the widely accepted axioms of set theory is the Zermelo-Fraenkel axioms with the axiom of choice. But the last axiom seems…
bjorn
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Advantage of accepting the axiom of choice

What is the advantage of accepting the axiom of choice over other axioms (for e.g. axiom of determinacy)? It seems that there is no clear reason to prefer over other axioms.. Thanks for help.
user24796
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Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we bother with the extra complications in Artin's…
Mathemagician
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Do We Need the Axiom of Choice for Finite Sets?

So I am relatively familiar with the Axiom of Choice and a few of its equivalent forms (Zorn's Lemma, Surjective implies right invertible, etc.) but I have never actually taken a set theory course. I know there are times when an explicit way of…
nullUser
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If all sets were finite, how could the real numbers be defined?

An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who supported such a system. I can see that the natural…
AgCl
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Over ZF, does "every Hilbert space have a basis" imply AC?

I know there is a similar result due to Blass [1] that over ZF, "every vector space has a (Hamel) basis" implies AC. Looking around, however, I can't find any results on the question for Hilbert spaces. I also don't see how to generalize Blass's…
Kameryn Williams
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Why is the continuum hypothesis believed to be false by the majority of modern set theorists?

A quote from Enderton: One might well question whether there is any meaningful sense in which one can say that the continuum hypothesis is either true or false for the "real" sets. Among those set-theorists nowadays who feel that there is a…
Lavender Lullaby
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Is Banach-Alaoglu equivalent to AC?

The Banach-Alaoglu theorem is well-known. It states that the closed unit ball in the dual space of a normed space is $\text{wk}^*$-compact. The proof relies heavily on Tychonoff's theorem. As I have recently figured out thanks to the nice guys on…
JT_NL
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What was the definition of "set" that resulted in Russell's paradox?

Russell's paradox, the set of all sets not containing themselves can be broken down to two statements: A thing that contains all sets that don't contain themselves. This thing/one such thing would necessarily qualify as a set. Now, what was the…
user2268997
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Positive set theory, antifoundation, and the "co-Russell set"

A more focused version of this question has now been asked at MO. Tl;dr version: are there "reasonable" theories which prove/disprove "the set of all sets containing themselves, contains itself"? Inspired by this question, I'd like to ask a…
Noah Schweber
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Do the axioms of set theory actually define the notion of a set?

In Henning Makholm's answer to the question, When does the set enter set theory?, he states: In axiomatic set theory, the axioms themselves are the definition of the notion of a set: A set is whatever behaves like the axioms say sets behave. This…
justin
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Why is it considered unlikely that there could be a contradiction in ZF/ZFC?

EDIT: No answer addresses the "bottleneck" question. It's not surprising to me because the question is vague. But I would like to know whether that is indeed the reason, or perhaps something else. The question is interesting to me and I would be…
user23211
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Simple/natural questions in group theory whose answers depend on set theory

Take these two questions: "Given objects $X$, is there always a group $(X, e, *)$ with those objects?” (Ans: yes iff Axiom of Choice.) "Take a group $G$, its automorphism group ${\rm Aut}(G)$, the automorphism group of that ${\rm Aut}({\rm…
Peter Smith
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How much set theory does the category of sets remember?

Question. Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is $M$ determined by $\mathbf{Set}_M$ as a category up…
Zhen Lin
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Book on the Rigorous Foundations of Mathematics- Logic and Set Theory

I am asking for a book that develops the foundations of mathematics, up to the basic analysis (functions, real numbers etc.) in a very rigorous way, similar to Hilbert's program. Having read this question: " Where to begin with foundations of…
Nameless
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