# Chain condition in iterated forcing

(I apologize for the long question, which has no mathematical content. Just looking for the right reference.)

In their celebrated paper [ST1971] introducing iterated forcing, Solovay and Tennenbaum showed the following theorem:

Theorem A: If $(B_\alpha: \alpha< \omega_1)$ is an increasing sequence of complete subalgebras of the Boolean algebra $B_{\omega_1}$, and $\bigcup_{\alpha<\lambda} B_\alpha$ is dense in $B_\lambda$ for each limit ordinal $\lambda\le \omega_1$, and if all $B_\alpha$ ($\alpha<\omega_1$) satisfy the ccc (countable chain condition), then also $B_{\omega_1}$ will satisfy the ccc.

This theorem is used to show that the finite support iteration of ccc forcings is again ccc. (Theorem 6.3 in [ST1971])

An essential fragment of the proof given in the paper is due to Silver. (The authors write that Silver's version is "quite a bit simpler than [our] original proof".)

Essentially the same proof shows the following theorem:

Theorem B: Let $\kappa$ be regular uncountable. Let $(P_\alpha:\alpha <\kappa)$ be an iteration of forcing notions with direct limit $P_\kappa$, and assume that the set of stages $\delta<\kappa$ where $P_\delta$ is the direct limit of the previous forcings is stationary in $\kappa$. If all $P_\alpha$ satisfy the $\kappa$-cc, then so does $P_\kappa$.

(In Jech's book, Theorem A is 16.9/16.10, and Theorem B is 16.30. The latter theorem has a 3-line proof, starting with "Exactly as the proof of 16.9.")

Question: To whom should Theorem B be credited?

• To Solovay-Tennenbaum, whose unpublished original proof of Theorem A most likely also showed Theorem B?
• Or to Silver, whose proof of Theorem A definitely can be easily generalized to a proof of Theorem B?
• Or to "Silver/Solovay-Tennenbaum", or "Silver's proof in [ST]"?
• Or just to Solovay? Solovay's remarks in [K2011] indicate that the proof of the iteration theorem is his. But I think it is customary (at least in mathematics) not to divide credit between the coauthors of a paper, unless such a division is explicitly mentioned in the paper.
• Or to somebody else, who first explicitly formulated Theorem B?

(I am asking because I want to add a remark to a proof of a variant of Theorem B; the proof will be a variant of the S/S-T proof.)

[K2011]: Akihiro Kanamori: Historical remarks on Suslin's Problem, in: Juliette Kennedy and Roman Kossak, editors, Set Theory, Arithmetic and Foundations of Mathematics: Theorems, Philosophies, Lecture Notes in Logic, volume 36, 1-12. Association for Symbolic Logic, 2011. (MR2882649. http://math.bu.edu/people/aki/18.pdf )

[ST1971]: Solovay, R. M.; Tennenbaum, S. Iterated Cohen extensions and Souslin's problem. Ann. of Math. (2) 94 (1971), 201–245. (MR0294139, DOI:10.2307/1970860)

• In lieu of any other answer, follow Stigler's law of eponymy, and give the credit to Feynman. – Asaf Karagila Apr 12 '18 at 21:59