The RaisonnierShelah construction of a nonmeasurable set
Brian Semmes
In this paper 1 we will give a proof of the following theorem of S. Shelah.
Theorem 1:
In Zermelo Fraenkel set theory (ZF) plus the axiom of dependent choice (DC),
we can prove that there is a nonmeasurable set if there is an uncountable
wellordered set of reals.
Shelah's proof uses rather sophisticated metamathematical arguments that
may not be accessible to the general mathematician. As our principle goal
is to reach a relatively wide audience, we will use the ideas of J. Raisonnier,
who has given a simpler and less metamathematical proof of Theorem 1. However
we will not follow Raisonnier's proof exactly. We will make some
simplifications to Raisonnier's arguments and we will also follow to a certain
extent the exposition of Raisonnier's proof presented in M. Bekkali's lecture
notes for a seminar taught by S. Todorcevic.