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Results tagged with set-theory
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user 4213
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.
27
votes
Accepted
Cardinality of the permutations of an infinite set
$k^k$.
Easy that it's an upper bound. For lower bound split $X$ into two equinumerous
subsets; there are $\ge k^k$ permutations swapping the two subsets.
- 20.8k
9
votes
Axiom of Infinity needed in Cantor-Bernstein?
One of the standard proofs avoids the Axiom of Infinity.
It's based on the Tarski fixed-point theorem, see for instance
www.cs.ucla.edu/~palsberg/course/cs232/papers/bernstein.pdf .
But it does use th …
- 20.8k
5
votes
Accepted
Equality of two circular sets
In ZF minus the axiom of foundation there is no way of proving
that all "circular" sets are equal. You could take a model of set theory
allowing ur-elements and replace some or all of these by circula …
- 20.8k