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 (X<sup>I</sup>) (X,I epsilon M), we construct a normal filter <sup> Gamma</sup> of subgroups of a group of automorphisms of Aut(B ), the <sup>Gamma</sup>-stable subalgebra B<sup> Gamma</sup> of B, an automorphism of the replica B<sup>Gamma</sup> and B<sup>Gamma</sup> and, an ultrafilter U that in a natural sense is generic in B<sup> Gamma</sup>, so that pi induces an automorphism of m<sup>Gamma</sup>/U. Part of the construction is quite general and applies to any B = RO(X<sup> I</sup>). (Chapters I-IV.) In Chapter I, by simulating the construction of B = RO(X<sup>I</sup>) 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<sub>b</sub> = {<ŷ<sub> b</sub>,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<sup>Gamma</sup>/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 m<sup>B</sup> via locally expressible permutations pi (epsilon M) of the extension of I. If pi is locally-expressible, formulae of the form &phis; (pix,...,piX<sub> n</sub>), (X<sub>1</sub>,...,X<sub>n</sub> epsilon M, piM, pi epsilon M, can be considered as formulae of the language of M. In chapter V, we consider the m<sup>Gamma</sup>'s introduced previously with B=RO<sup>(2oxox(kappa+1)</sup>) kappa an o-non-standard number in m. Results from earlier chapters lead in each case to automorphisms pi of m<sup> Gamma</sup> and generic ultrafilters U, so that pi induces an automorphism of m<sup>Gamma</sup>/U.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:704589 |
| Date | January 1983 |
| Creators | Hernandez Manfredini, Enrique German |
| Publisher | Royal Holloway, University of London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://repository.royalholloway.ac.uk/items/7252e090-92b2-46c9-82e0-24a47e1dc9b1/1/ |
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