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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The proof of $\textbf{Lemma 10.1}$ in Nik’s book about forcing

I’m an undergraduate student trying to teach myself set-theory. And I have some trouble understanding the density of a constructed set. In Lemma 10.1 of Nik’s book, it states: Let $G$ be a generic ...
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Explaining Conditional Probability and the Urn problem with white and black balls

I am trying to interpret the meaning behind the conditional probabilities when setting up the problem. Can someone confirm with me if I am understanding the problem correctly? Problem Statement: There ...
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Examples of Substructures that "do not know they are that substructure"

Just learned $\mathbb{L}\vDash \mathbb{V}=\mathbb{L}$ and was warned that this property is not obvious with the counterexample mentioned being $HOD$. I can think of a few examples of definable ...
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Is the set of Turing Machines decidable in ZFC non-recursive?

Let S be the set of all the TMs which halting is decidable in ZFC (for each TM in S, we can find one algorithm in ZFC that determines whether the machine halts or not). Is S recursive? Is there one ...
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Clarification on Paragraph in 'Introduction to Set Theory' By J. Donald Monk

Background: I have just started reading J. Donald Monk's "Introduction to Set Theory," and I would like to double-check my understanding of the following paragraph: The fundamental idea in ...
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Letting $\{a_i\}$ and $\{b_i\}$ be families of cardinal numbers $a_i < b_i$, prove there is an injection between $\sum_{i}a_i$ to $\prod_i b_i$

Letting $\{a_i\}$ and $\{b_i\}$ be families of cardinal numbers $a_i < b_i$, prove there is an injection between $\sum_{i}a_i$ to $\prod_i b_i$. I need it to complete the solution to Halmos' ...
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Could we regard $\aleph_0\in \mathbb Z$? [closed]

I hope that $\aleph_0\in \mathbb Z$, however if so, $\aleph_1={\aleph_o}^{\aleph_0}\in\mathbb Z$ should also be true, since $\mathbb Z$ is closed for multiplication, but $\aleph_1$ is for uncountable ...
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Are Set theoretic constructions of mathematical objects unique

It is generally believed that mathematics (at least a considerable portion) can be embedded in ZFC set theory. But one would wonder what this statement really means, and whether the set theoretic ...
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the union of a chain of well ordered sets is well ordered

I have difficulty to understand the following claim in "Naive set theory" of Halmos on page 68. Let $\mathcal{C}$ be a continuation chain of well-ordered sets, and $U$ be the union of these ...
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Are the open sets of $\{0, 1\}^I$ measurable?
Let $I$ be an uncountable set and let $2^I=\{0, 1\}^I$ be the set of all functions from $I$ into $\{0, 1\}$. Consider the product measure $\mu$ on $2^I$. The domain of this measure is the $\sigma$-...