Baptist Faith And Message Study Guide Pdf
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The goal of this work is to define the full, non-trivial space of first-order theories of arithmetic, and to study these theories, both from the foundations of mathematics and from the perspective of the philosophy of mathematics. It is intended to be an accessible, introductory, comprehensive and yet accessible account of the formal theory of arithmetic, and of its connection with the study of the continuum, and the logico-mathematical foundation of these studies. It is written to be straightforward and non-technical, not 'abstract'.
We begin with the definition of arithmetic, and its explanatory principles. Next, we study the standard construction of the natural numbers, and define arithmetic in the usual way from it. We then give a modal characterisation of arithmetic, using a generalisation of Gödel's constructibility interpretation of arithmetical properties of numbers. We then turn to an analysis of the main axioms of arithmetic, with an emphasis on the formalism of arithmetic; this is followed by a study of the informal semantic content of these axioms, as a study of the informal semantics of the language of arithmetic. Next, we develop a proof-theoretic account of the formal theory of arithmetic, and its main proofs. Finally, we introduce the canonical model of arithmetic; this is an object of algebraic geometry, and is mostly of interest to mathematicians. We then give a detailed introduction to the theory of the real numbers, a study of the topology of the reals, and a detailed study of the real numbers as a field. Further, we apply these results to establish the real numbers as a field.
This work has been co-authored with Neil Tennant and with Martin Loeb. We are indebted to the following for help and suggestions: Martin Dummett, William David Lewis, and Tim Button; to Gabriele D'Annibale, Marcello D'Ariano, Edward Nelson, and Toby Lindstrom; and to Mark Rogers, Daniele Zorzi, and the numerous students and post-docs with whom we have interacted and studied.
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