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Results tagged with set-theory
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user 497904
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
11
votes
2
answers
810
views
Undefinable inner model
What are some examples of a pair $M\subseteq N$ of transitive set models of $\mathsf{ZFC}$ with the same ordinals, such that $M$ is not a definable class (with parameters) in $N$? Is it possible that …
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7
votes
0
answers
374
views
Is Borel cardinality the same as cardinality under determinacy?
Suppose $E,F$ are Borel equivalence relations on Polish spaces $X,Y$, respectively. Under strong enough determinacy axioms, is it true that $E$ Borel reduces to $F$ iff there is an injective map from …
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6
votes
How much of the axiom of choice do you need in mathematics?
Each of the following is, at the very least, convenient for usual mathematics, but probably to a large extent unnecessary.
Countable/dependent choice. With them analysis and measure theory can be dev …
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7
votes
1
answer
609
views
What can be the measure of a Vitali set?
Suppose the continuum $\mathfrak{c}$ is real-valued measurable, i.e., there exists a countably additive probabilistic measure on $\mathfrak{c}$ that measures all subsets. Then by the construction on p …
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9
votes
0
answers
250
views
Naive way to violate $\mathsf{SCH}$ at $\aleph_\omega$
I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ a …
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7
votes
0
answers
290
views
Did Lebesgue like non-measurable set or not?
I was surprised by the following paragraph in Bressoud's A radical approach to Lebesgue's theory of integration, quoted by Caicedo's in his comment to this question:
Vitali's nonmeasurable set, appea …
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7
votes
1
answer
316
views
Strength of Borel determinacy
In this blog post by Gowers on Borel determinacy, Andres Caicedo says the following in a comment (slightly rephrased).
Let $\mathsf{ZFC^-}$ be $\mathsf{ZFC}$ without power set and $\mathsf{ZC^-}$ be …
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3
votes
0
answers
214
views
Basic cardinal arithmetic without choice
Do we know everything about addition and multiplication of cardinalities in choiceless set theory?
For example, let $M$ be a model of $\textsf{ZF}+\textsf{AD}+V=L(\mathbb{R})$, consider the sets $\mat …
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11
votes
3
answers
785
views
When are two forcing posets "the same"?
Let $B$ and $C$ be complete Boolean algebras. To avoid triviality I may also want them to be atomless. For $b\in B$ nonzero, denote $B\upharpoonright b=\{p\in B:p\leq b\}$, which can be viewed as a co …
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6
votes
Who needs Replacement anyway?
Isn't replacement needed or at least the most natural way to construct projective/injective resolutions? Say we want a free resolution of $M$. Consider the free module $F_1$ with $M$ as the set of gen …
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