Chapter

Incompleteness, Consistency, and the Ascent to Platonic Rationalism

Richard Tieszen

in After Gödel

Published in print May 2011 | ISBN: 9780199606207
Published online May 2011 | e-ISBN: 9780191725500 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199606207.003.0002
Incompleteness, Consistency, and the Ascent to Platonic Rationalism

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This chapter contains a brief overview of Gödel's main results on incompleteness and undecidability, consistency, finite axiomatizability, independence, and speed-up techniques. Tarski's indefinability theorem is also mentioned. Most of the chapter, however, is devoted to laying out the philosophical and foundational background against which these results emerged, including the ideas lying behind Hilbert's program, Carnap's early view of mathematics as syntax of language, and the perspective of intuitionism. The idea is to trace an ascent to a type of platonic rationalism about mathematics and logic, based on Gödel's own philosophical remarks on his results. Some texts that represent Gödel's earliest expressions of his platonism are cited. In later chapters of the book it is a defensible form of this platonic rationalism that will be developed in light of ideas in Plato, Leibniz, Kant, and Husserl.

Keywords: incompleteness; consistency; undecidability; indefinability; platonism; Hilbert; intuitionism; Carnap; set theory; truth

Chapter.  14150 words. 

Subjects: History of Western Philosophy

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