Large-Scale Formal Proof for the Working Mathematician ALEXANDRIA

Problem 11 is due to P. E. Howard.’ The results of Problems 12 and 13 were announced by Pincus. The independence result in Theorem 7.1 1 was proved by Levy [I9621 for ZFA and transferred to Z F by Pincus [ 1969,1973b1, whose is also the observation in Problem 15.’ Example 7.12 in Section 7.4 is due to Tarski. Conditions (S) and (D) were formulated by Mostowski [1945], who proved the independence result of Theorem 7.16 (for ZFA) and the sufficiency of Condition (D) (Problem 20); the result of Problem 16 is also his. The proof of the sufficiency of condition ( S ) given here is due to Szmielew [1947]. The independence result in Problem 21 was announced by Gauntt [1970]. 3 * 4 For additional results concerning the Axiom of Choice for finite sets, see the papers of Wiiniewski and Zuckerman. The results of Problems 23 and 24 are due to Morris [1969].5 The independence of the Prime Ideal Theorem from the Hahn-Banach Theorem was proved by Pincus [1972b]. The independence of the Prime Ideal Theorem from the Order Extension Principle (Problem 27) is due to Felgner [1971b].