Just a short post linking to my Part III Essay. It’s on the topic of Infinite Games and Large Cardinals. I’ll try to provide a high-level summary of the key concepts here.



Infinite Games

Infinite Games are, in this case, two-player games of perfect information which are played for \(\omega\) many turns. That is, there are a countably infinite number of moves made in the game, and only after a countably infinite number of moves is a winner determined. Each turn, a player plays a move consisting of a natural number. Every finite two-player game of perfect information can trivially be made into an infinite game by first coding every possible move with a number; then playing until the end of the finite game; and then each player can play any number for the remainder of the game; then whoever won the finite game wins the “infinite game”. A less trivial example might be the prime factor game where the rules are:

  • Player 1 plays any natural number at each turn.
  • Player 2 has to play every prime factor of numbers played by player 1.
  • Player 1 starts.

Can you determine who wins this game?

Answer Although it may seem that Player 1 can win by e.g. playing numbers with 2 distinct prime factors that have not yet been played by Player 2 at each turn, this does not give Player 1 a winning strategy! Since we play until infinity, player 2 will always eventually be able to "catch up" with Player 1 in the limit! So Player 2 has a winning strategy. What happens if Player 2 starts? Can Player 1 guarantee a win? How?



Large Cardinals

Large Cardinals are not just “really big” cardinals (they usually are also really big cardinals, but they aren’t just that). They’re cardinals which are so big that \(\mathrm{ZFC}\) cannot prove that they exist. This is because these cardinals are big enough to imply that \(\mathrm{ZFC}\) is consistent. This is because they will either be — or imply the existence of — a cardinal \(\kappa\) such that \(V_{\kappa}\) models \(\mathrm{ZFC}\) (if you’re confused by this \(V_{\kappa}\) notation, here you go). There are very many large cardinal hypotheses that one can accept, and they form something of a well-founded linear hierarchy (which is surprising! It’s known as the linearity phenomenon1). See a picture of (some of) this hierarchy here



My Essay

My essay explores the link between a certain weird type of large cardinal hypothesis2 called \(0^{\#}\) (or more accurately “\(0^{\#}\) exists”) and a determinacy axiom of Infinite Games. By a determinacy axiom for an infinite game, we mean something like “such-and-such a class of games is determined.” For example, one could suppose that every Infinite game is determined, and it seems that this is consistent with \(\mathrm{ZF}\) (but not consistent with the Axiom of Choice due to certain downstream implications about ultrafilters on \(\omega\)). In this essay, we don’t take such a strong determinacy condition, but rather take a determinacy condition on a class of games which are called “Analytic” games3. Then as the title of the essay — Large Cardinals from Determinacy — suggests, we go on to prove that this determinacy condition implies the existence of \(0^{\#}\). It turns out this is also an equivalence (but this was beyond the scope of my essay).



The Introduction from my Essay

In this essay, we prove a result of Harrington that analytic determinacy implies the existence of \(0^{\#}\) (zero sharp), which is a particular type of large cardinal hypothesis. To provide the necessary background for this proof, we will first outline some basic concepts and results in infinite games. Then in section two we will define analytic determinacy and provide the necessary context of Borel pointclasses, projective pointclasses, and the boldface and lightface hierarchies. We will also (very) briefly touch on some historical determinacy results. Section three will then serve to elucidate \(0^{\#}\). The presentation we will give will demonstrate how \(0^{\#}\) arose historically, as a construction flowing naturally from ideas in model theory, although towards the end of the section we will touch on a more contemporary perspective of \(0^{\#}\), which is how we will use it in the rest of the essay. Section four will outline a proof of Harrington’s theorem: that analytic determinacy implies the existence of \(0^{\#}\). The beginning of this proof will provide a proof of the existence of \(0^{\#}\) from a statement we will call Harrington’s principle. The remainder of section four will prove that analytic determinacy implies this principle and thus implies the existence of \(0^{\#}\). Finally, we will briefly discuss the consistency strength of analytic determinacy and \(0^{\#}\), as well as some related results, such as the Martin-Harrington theorem.




  1. Joel David Hamkins seems to think it’s not quite as simple as most set theorists like to think. ↩

  2. Weird because this hypothesis itself actually doesn’t make any claim about the existence of a particular large cardinal, and in fact is provably not even equivalent to the existence of a certain type of large cardinal. It does imply the existence of certain “weaker” large cardinals, but there is no specific large cardinal with the same consistency strength as the existence of \(0^{\#}\). ↩

  3. Explaining what an “analytic” game is in detail would defeat the aim of this high-level summary, but the essay awaits if you want to know more ;) ↩