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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.

1 vote
1 answer
723 views

Choice function on the countable subsets of the reals

It is easy to find a choice function on all finite subsets of $\mathbb R$, but without using the axiom of choice, not on all subsets. Is there an "explicit" choice function on the countable subsets of …
mathahada's user avatar
  • 656
12 votes
4 answers
1k views

Universal order type

Every countable order type, such as the countable ordinals, $\mathbb Z$, etc. can be embedded in $\mathbb Q$, so it is universal for countable order types. Is there a universal space for all linear or …
mathahada's user avatar
  • 656
2 votes
1 answer
1k views

Total order on the powerset

Given a well ordering of a set $A$ we can define a total order $A^A$ in an obvious way (for $f \neq g$ find the least $i$ such that $f(i) \neq g(i)$ and define $f < g$ if $f(i) < g(i)$) Does the inve …
mathahada's user avatar
  • 656
10 votes
2 answers
911 views

Set theoretic question about real valued functions

It's a question I've been thinking about but I can't find an easy answer. I think it will be interesting. Can there be a countable collection of real valued functions $f_1, f_2 , ... $ such that for a …
mathahada's user avatar
  • 656