Alice and Bob visit the cardinal, Part II

(Part I of this post can be found here.)

Greetings, loyal blog readers! I'm afraid life took over for a bit after writing Part I, but we're now back in the peanut gallery watching Alice and Bob battle wits.

In the last post, we talked about a fun game which Matt Baker used to prove the uncountability of the reals. (We'll call this the Nested-Intervals Game.) In his post about this game, he asked a question which (still weeks later) is vexing me:

Question 0: Does there exist a target set \(T\) such that neither player has a winning strategy for the Nested-Intervals Game targeting \(T\)?

Alice and Bob visit the cardinal, part I

(Part II of this post can be found here.)

Speaking of diagonal arguments: I ran across the blog of one Matt Baker yesterday, who sketched out probably the easiest proof I've ever seen of the uncountability of the real line. He also included a question in his post, one that I have a strong intuition about the answer, but so far haven't been able to prove I'm right.

(NB: when I say easiest proof I've ever seen, I mean that I sat down at lunch with a colleague who hadn't seen math since her freshman year of calc, and we finished lunch with her pretty much all over that shit.)

Anyway, I thought I'd record that argument in case Baker's blog disappears or (as has happened twice today) I can't figure out search terms to find it again. The next post will discuss his question, the version of an answer I can prove, and what makes the full problem more difficult. I'll try to pitch the level of these posts (well, more this one than the next) at the level of my lunch colleague.

A Left-for-the-reader in stability theory

Yes, yes, I know I haven't posted in forever. I've been busy proving shit.

Completely out of left field, my research has indicated relevance to stability theory, an area of model theory that I've never had the urge to learn. Well, now I feel the urge. So much for purity of intention.

Ralph and I had a little fun today thinking through a left-for-the-reader in a paper that I hope to raid for its methods. So much fun that I want to share it with you, O gentle reader.

First day of Mile High

Today was the first day of the Third Mile High Conference on Nonassociative Mathematics at University of Denver. Looks like a good group of people, only a few of whom I know. Yay for making connections.

Had lunch with Aleš Drápal today; I don't believe that we'd ever actually met before, though he immediately knew who I was, as I expected. The world is still small.

Neither he nor I are actively working on LD-related research at the moment, though finite LDs form nice test examples for computational packages; he mentioned that he has some unpublished theorems relating to the "levels" of homomorphisms between Laver tables. He also mentioned a direction of possible research that he had mostly abandoned, but where there might be results of manageable difficulty for a person or collaboration with the right crossover knowledge:

Let \( \mathbf{A} = \langle A; * \) be a LD-groupoid (probably finite and monogenerated). It is known that in some cases, such as when \( \mathbf{A} \) is a Laver Table, we can define an associative composition \( \circ \) on \( A \) satisfying

• \( ( a \circ b ) * c = a * (b * c) \)
• \( a \circ b = (a * b) \circ a \)
• \( a * (b \circ c) = ( a * b ) \circ (a * c) \)

and that this operation is provably unique in the case of Laver Tables. (If \( \mathbf{A} \) is a Laver Table, then \( (a \circ b)^+ = a * (b^+) \).) Aleš thinks that the semigroups obtainable in this way may be interesting from a semigroup theory perspective, but couldn't find any collaborators in the 90s whom he could interest in looking at the problem in any level of detail. (He thinks there's less there in terms of the universal algebraic properties of these semigroups -- what kind of varieties or quasivarieties they generate, blah blah blah...)

To future me, who wants to play around with this problem: start with Aleš' paper in Semigroup Forum 51, "On the semigroup structure of cyclic left distributive algebras".