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Title: Automorphisms of Boolean-value models of set-theory
Author: Hernandez Manfredini, Enrique German
Awarding Body: University of London
Current Institution: Royal Holloway, University of London
Date of Award: 1983
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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" : xˆb = {<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,...,Xn 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics