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| Paper IPM / M / 17375 |
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| Abstract: | |
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he proofs of Gèodel (1931), Rosser (1936), Kleene (first 1936 and second 1950), Chaitin (1970), and Boolos (1989) for the first incompleteness theorem are compared with each other, especially from the viewpoint of the second incompleteness theorem. It is shown that Gèodelâ??s (first incompleteness theorem) and Kleeneâ??s first theorems are equivalent with the second incompleteness theorem, Rosserâ??s and Kleeneâ??s second theorems do deliver the second incompleteness theorem, and Boolosâ?? theorem is derived from the second incompleteness theorem in the standard way. It is also shown that none of Rosserâ??s, Kleeneâ??s second or Boolosâ?? theorems is equivalent with the second incompleteness theorem, and Chaitinâ??s incompleteness theorem neither delivers nor is derived from the second incompleteness theorem. We compare (the strength of) these six proofs with one another. After discussing some other proofs of Gödel's First Incompleteness Theorem, namely the proofs of Rosser, Kleene, Chaitin and Boolos, we investigate whether these proofs of the first theorem also imply the second. It is shown that while the proofs of Rosser and Kleene imply the second theorem, the proofs of Chaitin and Boolos do not.
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