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Permanent link (DOI): https://doi.org/10.7939/R31V5BN67

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The Status of Mathematical Induction in an Axiomatic System Open Access

Descriptions

Other title
Subject/Keyword
Dedekind
Mathematical Induction
Frege
Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Sebti, Reza
Supervisor and department
Linsky, Bernard (Philosophy)
Examining committee member and department
Hazen, Allen (Philosophy)
Bulitko, Vadim (Computing Science)
Pelletier, Francis Jeffry (Philosophy)
Department
Department of Philosophy
Specialization

Date accepted
2014-09-10T16:02:42Z
Graduation date
2014-11
Degree
Master of Arts
Degree level
Master's
Abstract
This thesis investigates the status of Mathematical Induction (MI) in an axiomatic system. It first reviews and analyses the status of MI in the works of Gotlob Frege and Richard Dedekind, the pioneers of logicism who, in providing foundations for arithmetic, attempted to reduce MI to what they considered logic to be. These analyses reveal that their accounts of MI have the same structure and produce the same result. This is true even though the two thinkers used different components as fundamental logical elements and went through different routes to eventually prove (on the basis of more fundamental logical axioms and rules of inference and definitions) what they considered MI to be. Based on these analyses, we infer a formulation, i.e., U-MI, that presents both Frege’s and Dedekind’s formulations of MI. We then evaluate the possible proof- and model-theoretic problems that such a formulation of MI faces. These problems, among others, include certain difficulties with U-MI as a representation of mathematical induction, the problem of impredicativity, and the unattainability of the infinitary nature of MI in a finitary logic. We then introduce and defend our own account of the status of MI in an axiomatic system, in which MI is axiomatizable/derivable in an infinitary many-sorted logic. The final part of the study investigates concerns with the metatheoretical use of MI – in particular the circularity problem in such a use. Within this last part, we also explicate and elaborate on one of the advantages of our account of the status of MI in an axiomatic system in comparison to the rival accounts.
Language
English
DOI
doi:10.7939/R31V5BN67
Rights
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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