On counting problems in nonstandard models of Peano arithmetic with applications to groups

Reading, Alan G (2014). On counting problems in nonstandard models of Peano arithmetic with applications to groups. University of Birmingham. Ph.D.

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Abstract

Coding devices in Peano arithmetic (PA) allow complicated finite objects such as groups to be encoded in a model \(M\) ╞ PA. We call such coded objects \(M\) -finite. This thesis concerns \(M\) -finite abelian groups, and counting problems for \(M\)-finite groups. We define a notion of cardinality for non-\(M\) -finite sets via the suprema and infima of appropriate \(M\) -finite sets, if these agree we call the set \(M\) -countable. We investigate properties of \(M\) -countable sets and give examples which demonstrate marked differences to measure theory. Many of the pathologies are related to the arithmetic of cuts and we show what can be recovered in special cases. We propose a notion of measure that mimics the Carathéodory definition. We show that an \(M\) -countable subgroup of any \(M\) -finite group has an \(M\) -countable transversal of appropriate cardinality. We look at \(M\) -finite abelian groups. After discussing consequences of the basis theorem we concentrate on the case of a single \(M\) -finite group \(C\)(\(p\)\(^k\)) and investigate its external structure as an infinite abelian group. We prove that certain externally divisible subgroups of \(C\)(\(p\)\(^k\)) have \(M\) -countable complements. We generalize this result to show that \(d\)\(G\), the divisible part of \(G\), has an \(M\) -countable complement for a general \(M\) -finite abelian \(G\).

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

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