# 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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### Book Recommendation: Topology & Urelements [closed]

I am looking for a book about Topology with urelements. I am wondering whether the matter has ever been researched in the first place, especially since I could not find any trace on the internet: not ...
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### Kunen lemma 7.9, chapter IV.

In Kunens "Set Theory An Introduction to Independence Proofs", lemma 7.9, chapter IV, we are given the following: Let $G$ be an isomorphism from $A$ to $M$ which respects the $\in$-relation....
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### Constructive mathematics with different computational models

I am interested in understanding how the capabilities of constructive mathematics evolve when different computational models are considered. Specifically, if constructive mathematics traditionally ...
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### Countable Choice from Finite Sets

Consider the following 4 statements: Axiom of countable choice Axiom of countable choice from finite sets Axiom of countable choice from Dedekind finite sets Existence of a choice function for any ...
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### Motivation of inventing concept of well-ordered set?

I've started studying set theory for a while. I understand what is an ordered sets but i still fail to see what motivated mathematicians to invent these concept. Could you please enlightment me ? ...
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### Is there a model of ZF not C where not every set of reals is Lebesgue measurable? [duplicate]

I know that there is a model of ZF set theory plus the negation of the axiom of choice where every set of reals is Lebesgue measurable. But is there also a model where not every set of reals is ...
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### $\mathrm{Con}(\sf{ZFC}) \rightarrow \mathrm{Con}(\sf{ZFC}+\neg \exists \alpha(\text{$\alpha$is strongly inaccessible}))$

On page 133 in Kunens "Set Theory An Introduction to Independence Proofs", we are given the following (paraphrasing), If $\kappa$ is the first strong inaccessible, then $H(\kappa)$ is a ...
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### Principle in between BPI and AC

I have searched extensively in the literature, but all references I have consulted always place BPI (the Boolean Prime Ideal Theorem) as a sort of "cover" of the Axiom of Choice as far as ...
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### Greek letter "Kappa" use in Set Theory

I have been given a definition that the Greek letter "K" is used when a cardinal and ordinal number of a set are the same. As I have NO knowledge of Set Theory, can someone explain this? I ...
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### Inequality of cardinal sums [duplicate]

Define $|A|<|B|$ iff there is an injection from $A$ to $B$ but there is no injection from $B$ to $A$. Is it provable in ZF that $|A|<|C|$ and $|B|<|D|$ and $A\cap B=\emptyset=C\cap D$ implies ...
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### Epimorphism is equivalent to surjective using right-inverses.

I want to show that in the category of sets a function is an epimorphism iff it is surjective. The problem I'm having is with showing that epimorphism $\Longrightarrow$ surjective. I've seen the ...
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### Proof of Gödels first incompleteness theorem (as in Kunen)

On page 40 in Kunens "Set Theory An Introduction to Independence Proofs", we are given the following theorem: Gödel. If $\phi(x)$ is any formula with one free variable, $x$, then there is a ...
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### Absoluteness of ordinal exponentiation and rank on transitive models of $\sf{ZF}-\sf{P}$.

In Kunens "Set Theory An Introduction to Independence Proofs", page 129-130, we are given a proof that for any transitive model $\mathbf{M}$ of $\sf{ZF}-\sf{P}$, ordinal exponentiation and ...
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### Is there something missing in Jech's proof of Zermelo's Well-Ordering Theorem?

Here is the proof from p. 48 of the Millennium Edition, corrected 4th printing 2006: My question: how do we know that there is any ordinal $\theta$ such that $A=\{a_\xi\,\colon\xi<\alpha\}$? ...
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### Absoluteness of inaccessible cardinals

I'm studying large cardinals and I'm hoping to fully understand the proof that says ZFC is not able to prove the existence of inaccessibles (given ZFC is consistent, of course). I've already fully ...
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### Why are extensions of countable models of ZFC better behaved than extensions of arbitrary models of ZFC?

This answer hints that certain kinds of extensions are only guaranteed to exist for countable models of ZFC. Why? One intuitive reason i can think of is that the metatheory might not have enough new ...
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### What exactly are characteristic functions of sets when constructing Boolean models?

Reading J. L. Bell's Set Theory --- Boolean-Valued Models and Independence Proofs. This is how the introduction of Boolean models is motivated: Suppose that for each set $x ∈ V$ we are given a ...
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### Need to check this proof that the class of models of ZC that fail replacement is not axiomatisable.

Here ZC is ZFC minus Axiom of Replacement. My proof is as follows: Suppose $M$ was axiomatized by a theory $H$. For non-zero limit ordinal $\alpha$, let $T_\alpha$ be the set of the replacement axioms ...
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In this answer, Asaf Karagila says that it is consistent with ZFC that $\frak c=\aleph_\alpha$ for any ordinal $\alpha$ without cofinality $\omega$. It is not clear to me what exactly this means. If $\... • 457 13 votes 5 answers 3k views ### How do we define addition? I've been trying to learn naive set theory through a YouTube series by 'The Bright Side of Mathematics'. So far, I've been able to understand successor maps and the definition of$\mathbb{N}_{0}$. I ... 2 votes 1 answer 14 views ### Formalizing the construction of a Cantor scheme I'm asking this question here since a similar question asked on math.stackexchange didn't receive an answer. I'm trying to understand the proof of the following theorem: Theorem (Brouwer): The Cantor ... • 1,209 1 vote 0 answers 84 views ### Asking the experts in mathematical logic and set theory for survey articles to supplement the standard textbooks..... I'm asking this question to the experts in mathematical logic and set theory.I'm a former graduate student in mathematics with an additional background in philosophy engaged in serious self study. I'... • 17.4k 1 vote 1 answer 86 views ### Theorem 5.6, chapter IV in Kunens Set Theory Boldface-letters (i.e.$\mathbf{A}$,$\mathbf{V}$,etc.) indicate a class. The following is an excerpt from Kunens "Set Theory An Introduction to Independence Proofs" (theorem 5.6,$\S5$of ... • 1,308 2 votes 0 answers 133 views ### Why is the Axiom of Choice Necessary in ZFC Within the framework of Zermelo-Fraenkel set theory with the Axiom of Choice$(ZFC)$, when we considered the method of constructing the set of natural numbers, we regarded it as the smallest inductive ... • 115 0 votes 1 answer 30 views ### Infinite Ramsey Theorem analogue for infinite complete Bipartite Graphs The version of the Infinite Ramsey Theorem I am interested in states that for any countably infinite complete graph$K_{\aleph_0}$edge-coloured by$f: E(K_{\aleph_0}) \to [1, c]$(with$c \in \mathbb{...
In Kunens "Set Theory An Introduction to Independence Proofs", p. 128, theorem 5.5 in chapter IV, he wants to prove the following: For ordinals $\alpha$ and $\beta$, the following statement ...