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An informal exposition of proofs of Gödel's theorems and Church's theorem

  • Barkley Rosser (a1)
Abstract

This paper is an attempt to explain as non-technically as possible the principles and devices used in the various proofs of Gödel's Theorems and Church's Theorem.

Roman numerals in references shall refer to the papers in the bibliography. In the statements of Gödel's Theorems and Church's Theorem, we will employ the phrase “for suitable L.” The hidden assumptions which we denote by this phrase have never been put down explicitly in a form intelligible to the average reader. The necessity for thus formulating them has commonly been avoided by proving the theorems for special logics and then remarking that the proofs can be extended to other logics. Hence the conditions necessary for the proofs of Gödel's Theorems and Church's Theorem are at present very indefinite as far as the average reader is concerned. To partly clarify this situation, we will now mention the more prominent of these assumptions.

I. In any proof of Gödel's Theorems or Church's Theorem, two logics are concerned. One serves as the “logic of ordinary discourse” in which the proof is carried out, and the other is a formal logic, L, about which the theorem is proved. The first logic may or may not be formal. However L must be formal. Among other things, this implies that the propositions of L are formulas built according to certain rules of structure.

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References
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I.Church Alonzo, An unsolvable problem of elementary number theory, American journal of mathematics, vol. 58 (1936), pp. 345363.
II.Gödel Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173198. (A less difficult exposition of Gödel's work is to be found in Carnap's The logical syntax of language.)
III.Kleene S. C., General recursive functions of natural numbers, Mathematische Annalen, vol. 112 (1936), pp. 727742.
IV.Rosser Barkley, Extensions of some theorems of Gödel and Church, this Journal, vol. 1 (1936), pp. 8791.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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