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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Kunen "Set Theory" 2011 versus 1980 edition - worth buying again?

What are the differences between the original edition (1980) of Kunen's famous book and the new edition (2011)? Is the updated version worth buying? (I hope this kind of question is allowed here. I could not find a definitive answer in the…
Justus87
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Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?

Often, we use a construction like this: Given a subset $ A $ of a metric space and its limit point $ a $, we know that for every $ \epsilon > 0 $ there is another point $ x $ different from $ a $ such that $ d(x,a) < \epsilon $. So for every…
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Supercompact cardinals and being witnessed by a structure of limited rank

Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-huge cardinal) that is witnessed by a structure of limited rank exists above a supercompact cardinal κ, then a cardinal with that property exists…
Hello
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Size of a natural transformation and the Yoneda Lemma

Appearing on the second page (under the section Digression: Size worries) of the following paper about the Yoneda Lemma: http://www.maths.ed.ac.uk/~tl/categories/yoneda.ps It says that $a$ $priori$ $[\mathcal C^{op},Set](H_A,X))$ is a class. I…
user52534
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Does the Laver real determine the generic filter?

Let us concern the Laver forcing $ \mathbb{L} $. Let $ G $ be $ \mathbb{L} $-generic over a c.t.m. $ M $ for ZFC. Let $$ x_G := \bigcup \{ \operatorname{stem}(p) : p \in G \} $$ be the Laver real determined by $ G $. Now, Jech ("Set Theory", 3rd…
Justus87
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Intuition for "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
qazwsx
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$\{S\} \not\in S$ in ZFC?

The usual argument I see for proving that, assuming S is a set, $S\not\in S$ (in ZFC) is the following: Take $S$. Assume $S$ is a set in ZFC. Define $T=\{S\}$ ; by axiom of pairing, $T$ is a set. Thus it satisfies the axiom of regularity,…
JustAskin
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Constructively generating a sigma algebra

We have a collection $\mathcal{C}$ of sets (includes $\Omega)$ and would like to constructively generate the sigma algebra $\sigma(\mathcal{C})$. Would the following process work? Let $\mathcal{S}=\mathcal{C}$ 1.Take the complement of each set in…
iMath
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Where in the analytic hierarchy does V=L start having consequences?

I note that the ordinals of L are the same as V, so I guess that it has no $\Pi_1^1$ consequences. On the other hand Wikipedia tells me that it asserts the existance of a $\Delta_2^1$ non-measurable set of reals. "Measurable" involves third-order…
anonymous
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Levy collapse gone bad

Let $\kappa$ be strongly inaccessible, and let $\mu<\kappa$ be regular. What is the effect on $\kappa$ of forcing with the following? (1) The product of $Col(\mu,\alpha)$ for $\alpha<\kappa$, with supports of size $<\kappa$. (bounded support) (2)…
mbsq
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All models of $\mathsf{ZFC}$ between $V$ and $V[G]$ are generic extensions of $V$

I'm reading the proof of lemma 15.43 of Jech's Set Theory: Let $G$ be generic on a complete Boolean algebra $B$. If $M$ is a model of $\mathsf{ZFC}$ such that $V\subset M\subset V[G]$, then there exists a complete subalgebra $D\subset B$ such that…
Camilo Arosemena-Serrato
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Problem 24 from Chapter 1 of Kunen's Set Theory: An Introduction to Independence Proofs

Just want to make sure I'm tracking Kunen here, and hopefully the proof I have is correct. Comments / Suggestions welcome. Thanks! Problem 24. Let T be any consistent set of axioms extending ZF. Show that the Z = {$\psi$ : T $\vdash \psi$} is not…
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Exam question about stationary sets

This is an old exam question that I don't have an answer to. I am posting my solution in hope that if I have errors, someone will point them out, and for future generations of students taking this course. Let $\{A_\alpha \mid \alpha < \aleph_1\}$,…
Hila
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Can the power set be axiomatised?

I want to consider many-sorted first order logic with distinguished sorts $U$ and $P$. Can I state a (finite?) set of first order formulae such that any model $M = (D^U, D^P, I)$ interprets the sort $P$ as the set of finite subsets of the…
Matt
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What is a Cantor-style proof of $2^n > n^k$?

Cantor's diagonalization argument shows that no function from an $n$-element set to its set of subsets hits every element. This is one way to see that $2^n > n$ for every $n$. The classic application is when $n$ is an infinite cardinal, but in…
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