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Results tagged with set-theory
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user 1004
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.
23
votes
Accepted
Can every uncountable subset $\mathbb{R}$ be split at some number into two parts of the same...
No, the statement cannot be proven in ZFC without assuming continuum hypothesis or something similar. In fact, it is equivalent to the statement that there are finitely many cardinalities between $\al …
- 17.1k
18
votes
Accepted
Order type of the smallest set containing the identity function and closed under exponentiation
As Joel showed, the set $E$ is well-ordered with order type no more than the Cantor ordinal $\epsilon_0$. In fact, its order type is exactly $\epsilon_0$. This can be proved by constructing the order …
- 17.1k
26
votes
Non-Borel sets without axiom of choice
Measure theory without the Axiom of Choice (not even countable choice) is discussed in Fremlin, Measure Theory, Volume 5, Chapter 56. This is freely available online. Thanks to MO and ex-falso-quodlib …
- 17.1k