Some applications of extensions of ZF set theory (Q579248)

This is an expository paper containing sketches of proofs concerning three mathematical problems and their relation to various extensions of Zermelo-Fraenkel set theory: (1) Whitehead's Problem in group theory and its connection with V\(=L\) and Martin's Axiom (results of Shelah); (2) The existence of incomparable degrees of unsolvability above a given degree in ordinal recursion theory and its connection with \(V=L\) (result of S. D. Friedman); (3) The consistency of the Lebesgue measurability of all sets of reals and its connection with the consistency of the existence of inaccessible cardinals (results of Solovay and Shelah), with some simplifications based on work of Talagrand and Raisonnier).