Reflections on the Diagonal Theorem, and Related Topics

概要

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A formula (function), that expresses a diagonal sequence generated by all finite length formulas (functions), is an essentially infinite length formula (function). The last report made it clear that the substitution function and the provability predicate in Godel's paper are essentially infinitely long expressions. These indicate that Godel's proof of the incompleteness theorems is incorrect. On the same basis this report shows that the diagonal theorem does not hold. Since Godel's incompleteness theorems, Rosser's theorem, Tarski's theorem and the Π 1-incompleteness theorem are proved using the diagonal theorem. This does not mean that the theorems do not exist, but they must be reexamined. Furthermore, the recursion theorem and the fixed point theorem must be reviewed, because that these theorems are proved using essentially infinite length functions.
2011-05-09