Title:

Automorphisms of Booleanvalue models of settheory

This thesis is concerned with models m of ZF that admit automorphisms of order greater than 1. We obtain such models using Booleanvalued models. Starting with a fixed ononstandard countable m, and considering the algebra B epsilon M whose universe is B = RO (X^{I}) (X,I epsilon M), we construct a normal filter ^{ Gamma} of subgroups of a group of automorphisms of Aut(B ), the ^{Gamma}stable subalgebra B^{ Gamma} of B, an automorphism of the replica B^{Gamma} and B^{Gamma} and, an ultrafilter U that in a natural sense is generic in B^{ Gamma}, so that pi induces an automorphism of m^{Gamma}/U. Part of the construction is quite general and applies to any B = RO(X^{ I}). (Chapters IIV.) In Chapter I, by simulating the construction of B = RO(X^{I}) outside the model, we obtain a Booleanalgebra that is isomorphic to B. In Chapter II we list some known connections between generic ultrafilters and models of ZF which hold when m is nonstandard and B is replaced by B. We introduce the concept of mstandardness. In Chapter III the concepts of 'extendability', of 'almost genericity' and of 'locallyexpressible' permutations and automorphisms are introduced. A generalised version of the "xˆ's" : xˆ_{b} = {<ŷ_{ b},b>: y epsilon x} is given (x epsilon M, b epsilon B). Some of their properties are examined. It is shown that the condition pi[U] = U (*) is necessary and sufficient in order to induce automorphisms in m^{Gamma}/U, and that extendability constitutes a sufficient condition in order to obtain pi, U satisfying (*). Such pi,U are constructed simultaneously. In Chapter IV we construct automorphisms of two symmetric Booleanvalued submodels of m^{B} via locally expressible permutations pi (epsilon M) of the extension of I. If pi is locallyexpressible, formulae of the form &phis; (pix,...,piX_{ n}), (X_{1},...,X_{n} epsilon M, piM, pi epsilon M, can be considered as formulae of the language of M. In chapter V, we consider the m^{Gamma}'s introduced previously with B=RO^{(2oxox(kappa+1)}) kappa an ononstandard number in m. Results from earlier chapters lead in each case to automorphisms pi of m^{ Gamma} and generic ultrafilters U, so that pi induces an automorphism of m^{Gamma}/U.
