# Earliest proof of Solovay's theorem for successor cardinals

Solovay's partition theorem states that a stationary set over a regular cardinal $$\kappa$$ can be partitioned into $$\kappa$$-many disjoint stationary sets. The full theorem was proven by Solovay in 1971 (Robert M Solovay. Real-valued measurable cardinals), but the case of of successor cardinals is much simpler and can be proven using Ulam matrices.

I have seen this special case credited to Ulam in some places, I couldn't find any references of him actually proving it. The earliest proof I could find is by Fodor in 1966 (G fodor, On stationary sets and regressive functions in Szeged), while Ulam matrices were first introduced in 1930 with no mention of stationary sets.

I want to find the first proof using Ulam matrices, and verify whether or not Fodor's proof is the earliest one overall.

Thanks for any help!

• Stevo Todorcevic once told me that it was Ulam. Commented Jun 5 at 16:27
• Ulam's proof works for any kappa-complete ideal over kappa, be it NS_kappa or not. The only drawback is that Ulam's theorem is limited to kappa a successor. You could read more about it in the introduction to our paper "Was Ulam right? II" or your could watch the recording of the following lecture: youtu.be/xsVz4t8DB74
– saf
Commented Jun 5 at 16:56
• @JoelDavidHamkins I think I'll message him and see if he has any details. I'm writing a wikipedia page for Solovay's theorem, and I don't think "Joel David Hamkins, personal correspondence with Stevo Todorcevic" is a reliable reference :) Commented Jun 6 at 18:38
• @YnirPaz Good idea. I had the impression that Stevo often took it upon himself to say something when people would mistakenly credit the full result to Solovay, his point being that Ulam already had a good part of it. Commented Jun 6 at 18:47

Ulam matrices appeared in Ulam's doctoral thesis, written in Polish and presented at the Technical University of Lwów in 1933. The results of the dissertation were published in two articles [Fund. Math. 14 (1929), 231–233, and Fund. Math. 16 (1930), 140–150]. The first one is written in English. The second one is written in German, and it is the one in which Ulam matrices appear; a free copy of it can be found here.

Ulam matrices are used to prove the theorem below, which as explained by the remark following it, implies that every stationary subset of a successor cardinal $$\kappa$$ can be partitioned into $$\kappa$$ disjoint stationary sets.

The theorem and the remark following it are taken from Kunen's textbook "Set Theory" (1980 edition, p.79).

Theorem (Ulam). If $$\kappa$$ is a successor infinite cardinal, and $$\mathcal{I}$$ is a $$\kappa$$-complete ideal on $$\kappa$$ such that each singleton is in $$\mathcal{I}$$, then there are disjoint $$X_{\alpha}$$ for $$\alpha<\kappa$$ such that $$X_\alpha\notin \mathcal{I}$$ for each $$\alpha$$.

Remark. If $$\kappa$$ is as above, then whenever $$S$$ is any stationary subset of $$\kappa$$, $$S$$ maybe partitioned into $$\kappa$$ disjoint stationary subsets since we can apply Ulam's above theorem to the ideal $$\mathcal{I}$$, where $$\mathcal{I} =\{X \subseteq \kappa$$ : $$X \cap S$$ is nonstationary}. $$\mathcal{I}$$ is $$\kappa$$-complete since the intersection of less than $$\kappa$$-many closed unbounded sets is closed unbounded.

History of the concept of stationarity. According to Thomas Jech's textbook "Set Theory" (Millenium edition, p. 105) and this article by Jean Larson (footnote 213 on page 82) the notion of stationarity was explicitly introduced by Gérard Bloch in 1955, but the concept is implicit already in the work of Paul Mahlo (1911).

Two relevant MO answers are this MO-answer by Péter Komjáth and This MO answer by Joel David Hamkins.

• I found these, but they don't seem to mention anything about stationary sets (at least from the German I could read). I want to know if Ulam proved this himself before Fodor. Commented Jun 6 at 18:31
• @YnirPaz I have updated my answer with an addendum in light of your comment. Commented Jun 6 at 19:16