Interesting Esoterica

Gödel for Goldilocks: A Rigorous, Streamlined Proof of (a variant of) Gödel's First Incompleteness Theorem

Article by Dan Gusfield
  • Published in 2014
  • Added on
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Most discussions of G\"odel's theorems fall into one of two types: either they emphasize perceived philosophical, cultural "meanings" of the theorems, and perhaps sketch some of the ideas of the proofs, usually relating G\"odel's proofs to riddles and paradoxes, but do not attempt to present rigorous, complete proofs; or they do present rigorous proofs, but in the traditional style of mathematical logic, with all of its heavy notation and difficult definitions, and technical issues which reflect G\"odel's original approach and broader logical issues. Many non-specialists are frustrated by these two extreme types of expositions and want a complete, rigorous proof that they can understand. Such an exposition is possible, because many people have realized that variants of G\"odel's first incompleteness theorem can be rigorously proved by a simpler middle approach, avoiding philosophical discussions and hand-waiving at one extreme; and also avoiding the heavy machinery of traditional mathematical logic, and many of the harder detail's of G\"odel's original proof, at the other extreme. This is the just-right Goldilocks approach. In this exposition we give a short, self-contained Goldilocks exposition of G\"odel's first theorem, aimed at a broad, undergraduate audience.

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BibTeX entry

@article{GdelforGoldilocksARigorousStreamlinedProofofavariantofGdelsFirstIncompletenessTheorem,
	title = {G{\"{o}}del for Goldilocks: A Rigorous, Streamlined Proof of (a variant of)  G{\"{o}}del's First Incompleteness Theorem},
	abstract = {Most discussions of G\"odel's theorems fall into one of two types: either
they emphasize perceived philosophical, cultural "meanings" of the theorems,
and perhaps sketch some of the ideas of the proofs, usually relating G\"odel's
proofs to riddles and paradoxes, but do not attempt to present rigorous,
complete proofs; or they do present rigorous proofs, but in the traditional
style of mathematical logic, with all of its heavy notation and difficult
definitions, and technical issues which reflect G\"odel's original approach and
broader logical issues. Many non-specialists are frustrated by these two
extreme types of expositions and want a complete, rigorous proof that they can
understand. Such an exposition is possible, because many people have realized
that variants of G\"odel's first incompleteness theorem can be rigorously
proved by a simpler middle approach, avoiding philosophical discussions and
hand-waiving at one extreme; and also avoiding the heavy machinery of
traditional mathematical logic, and many of the harder detail's of G\"odel's
original proof, at the other extreme. This is the just-right Goldilocks
approach. In this exposition we give a short, self-contained Goldilocks
exposition of G\"odel's first theorem, aimed at a broad, undergraduate
audience.},
	url = {http://arxiv.org/abs/1409.5944v3 http://arxiv.org/pdf/1409.5944v3},
	year = 2014,
	author = {Dan Gusfield},
	comment = {},
	urldate = {2021-04-10},
	archivePrefix = {arXiv},
	eprint = {1409.5944},
	primaryClass = {math.LO},
	collections = {about-proof}
}