The foundations of mathematics include mathematical logic, set theory, recursion theory, model theory, and Gödel's incompleteness theorems. Professor Wolf provides here a guide that any interested reader with some post-calculus experience in mathematics can read, enjoy, and learn from. It could also serve as a textbook for courses in the foundations of mathematics, at the undergraduate or graduate level. The book is deliberately less structured and more user-friendly than standard texts on foundations, so will also be attractive to those outside (...) the classroom environment wanting to learn about the subject. (shrink)

We prove (in ZFC Set Theory) that all infinite games whose winning sets are of the following forms are determined: (1) (A - S) ∪ B, where A is $\Pi^0_2, \bar\bar{S}, 2^{\aleph_0}$ , and the games whose winning set is B is "strongly determined" (meaning that all of its subgames are determined). (2) A Boolean combination of Σ 0 2 sets and sets smaller than the continuum. This also enables us to show that strong determinateness is not preserved under complementation, (...) improving a result of Morton Davis which required the continuum hypothesis to prove this fact. Various open questions related to the above results are discussed. Our main conjecture is that (2) above remains true when "Σ 0 2 " is replaced by "Borel". (shrink)

This text is designed to teach students how to read and write proofs in mathematics and to acquaint them with how mathematicians investigate problems and formulate conjecture.