a060601 Astraendo
Topic
Peano Arithmetic
What It Is
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a060601 Astraendo
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{Ed.Note: This is the place for a basic summary of what Peano Arithmetic and the basic formulation that will be used. There is the informal notion, then the FOL= formulation. We think that PAO will be used here, but we need to be certain about that. Additional pages will have more detail. This page should serve as the latest overview and basic formulation.}
Preliminaries
Peano Arithmetic with Induction: PAI
Peano Arithmetic
- Boolos, George S., Burgess, John P., Jeffrey, Richard C. (2002)
- Computability and Logic, ed.4. Cambridge University Press (Cambridge & New York: 2002). ISBN 0-521-00758-5 pbk.
- Kennedy, Hubert C. (1973)
- Selected works of Giuseppe Peano. Translated and edited, with a biographical sketch and bibliography by Hubert C. Kennedy. University of Toronto Press (Toronto & Buffalo, NY: 1973); George Allen & Unwin (London: 1973). ISBN 0-04-164002-0.
Chapter VII, pp. 101-134, provides a translation of (Peano 1889) in its entirety, including Peano's selection of theorems and approach to presentation of proofs.
- Nelson, Edward (1986).
- Predicative Arithmetic. Mathematical Notes 32. Princeton University Press (Princeton, NJ: 1986). ISBN 0-691-08455-6 pbk.
- Peano, Giuseppe (1889).
- Arithmetices principia, nova methoda exposita. Bocca (Torino: 1889).
Written in a form of Latin that Peano sought to use as a universal language for exposition of mathematics, The principles of arithmetic, presented by a new method was the first effort to characterize arithmetic in an axiomatic symbolic logic. Translations appear in (Kennedy 1973; van Heijenoort 1977).
- van Heijenoort, Jean (ed.) (1977).
- van Heijenoort, Jean (ed). From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press (Cambridge, MA: 1967), 3rd (1977) printing. ISBN 0-674-32449-8 pbk.
pp. 81-97 provides an abridged translation of (Peano 1889) prepared by the editor. The first three pages include introductory remarks that provide some context and also tend to offer some critical hindsight where Peano did not meet the later standards of formal deductive logic. Nevertheless, Peano and Frege are credited with setting the stage for the development of Principia Mathematica, published in 1910. Many of Peano's notational ideas have survived in present-day mathematical logic, as has Peano Arithmetic itself.
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created 2006-06-08-14:18 -0700 (pdt) by orcmid |