Some Cardinal Arithmetic

Some time ago, Asaf Karagila wrote wonderful post wherein he shows that, even without assuming the axiom of choice one can always find four cardinals \(\mathfrak{p} \lt \mathfrak{q}\) and \(\mathfrak{r} \lt \mathfrak{s}\) such that \(\mathfrak{p}^{\mathfrak{r}} = \mathfrak{q}^{\mathfrak{s}}.\) In the comments, Harvey Friedman asks:

Your theorem is an example of an existential sentence about cardinals in the language with only \(\lt \) and exponentiation. Can you determine which sentences in that language are provable in ZF? More generally, expand the set of sentences about cardinals considered, obviously to include addition and multiplication, and perhaps alternating quantifiers.

After Peter Krautzberger’s tiny blogging challenge, I decided to spend 30 minutes thinking and writing about this…

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Diaconescu’s Theorem

In the 1970s, Radu Diaconescu [1] showed that the Axiom of Choice implies the Principle of Excluded Middle. More specifically, he showed that every topos that satisfies (a very mild form of) the Axiom of Choice is Boolean. Diaconescu’s argument applies in many other contexts too. In this post, I will present a variant of that argument. I will keep the context of the argument deliberately ambiguous to demonstrate its broad applicability.

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XKCD on the Axiom of Choice…

Arguably one of the best XKCD comics ever…

XKCD - 982 - Set Theory

The alternate text for this comic is a modern lumberjack’s proof of Zermelo’s Theorem.

Proof of Zermelo’s well-ordering theorem given the Axiom of Choice:
1: Take S to be any set.
2: When I reach step three, if S hasn’t managed to find a well-ordering relation for itself, I’ll feed it into this wood chipper.
3: Hey, look, S is well-ordered.