my forty years on his shoulders - Department of Mathematics

52iv. Consistency of the CH. Consistency of the most basic set theoreticmathematical problem highlighted by Cantor. (Gödel 1940).v. ∈ 0 consistency proof. Consistency proof of PA using quantifier freereasoning on the fundamental combinatorial structure, ∈ 0 . (Gentzen 1969).vi. Functional recursion consistency proof. Consistency proof of PA usinghigher type primitive recursion, without quantifiers. (Gödel 1958), (Gödel 1972).vii. Independence of AxC. Independence of CH (over AxC). Complementsiii,iv. (Cohen 1963-1964). Forcing.viii. Open set theoretic problems in core areas shown independent.Starting soon after (Cohen 1963-1964), starting dramatically with R.M. Solovay(e.g., his work on Lebesgue measurability (Solovay 1970), and his independenceproof of Kaplansky’s Conjecture (Dales, Woodin 1987)), and continuing withmany others. See the rather comprehensive (Jech 2006). Also see the many settheory papers in (Shelah, 1969-2007).Core mathematicians have learned to avoid raising new set theoreticproblems, and the area is greatly mined.ix. Large cardinals necessarily used to prove independent set theoreticstatements. Starting dramatically with measurable cardinals implies V ≠ L (Scott1961). Continuing with solutions to open problems in the theory of projectivesets (using large cardinals), culminating with the proof of projectivedeterminacy, (Martin, Steel 1989).x. Large cardinals necessarily used to prove the consistency of set theoreticstatements. See (Jech 2006).xi. Uncountably many iterations of the power set operation necessarily