Automorphisms of Boolean-value models of set-theory

Abstract

This thesis is concerned with models m of ZF that admit automorphisms of order greater than 1.

We obtain such models using Boolean-valued models.

Starting with a fixed o-non-standard countable m, and considering the algebra B epsilon M whose universe is B = RO (XI) (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 BGamma and BGamma and, an ultrafilter U that in a natural sense is generic in B Gamma, so that pi induces an automorphism of mGamma/U.

Part of the construction is quite general and applies to any B = RO(X I). (Chapters I-IV.)

In Chapter I, by simulating the construction of B = RO(XI) outside the model, we obtain a Boolean-algebra 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 non-standard and B is replaced by B.

We introduce the concept of m-standardness.

In Chapter III the concepts of 'extendability', of 'almost- genericity' and of 'locally-expressible' 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 mGamma/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 Boolean-valued submodels of mB via locally expressible permutations pi (epsilon M) of the extension of I.

If pi is locally-expressible, formulae of the form &phis; (pix,...,piX _n), (X₁,...,X_n epsilon M, piM, pi epsilon M, can be considered as formulae of the language of M.

In chapter V, we consider the mGamma's introduced previously with B=RO(2oxox(kappa+1)) kappa an o-non-standard 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 mGamma/U