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Gödel's second incompleteness theorem shows that, under general assumptions, this canonical consistency statement Cons(F) will not be provable in F. The theorem first appeared as "Theorem XI" in Gödel's 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I".
In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This weaker form differs from the standard statement of the incompleteness theorem by asserting that an axiomatization of the natural numbers that is both complete and sound is impossible.
Gödel's theorem applies to any formal theory that satisfies certain properties. Each formal theory has a signature that specifies the nonlogical symbols in the language of the theory. For simplicity, we will assume that the language of the theory is composed from the following collection of 15 (and only 15) symbols: A constant symbol 0 for zero.
Informally, the sentence employed to prove Gödel's first incompleteness theorem says "This statement is not provable." The fact that such self-reference can be expressed within arithmetic was not known until Gödel's paper appeared; independent work of Alfred Tarski on his indefinability theorem was conducted around the same time but not ...
In mathematical logic, Rosser's trick is a method for proving a variant of Gödel's incompleteness theorems not relying on the assumption that the theory being considered is ω-consistent (Smorynski 1977, p. 840; Mendelson 1977, p. 160).
The problem is that it is stated umpteen times in the literature that Hilbert was too thick to understand Godel's theorem, and that his response was uncomprehending. But this quote shows that he understood Godel's theorem fully, and was able to see how to extend the program in the way that it was later extended by Gentzen and others.
Gödel's original proof of the theorem proceeded by reducing the problem to a special case for formulas in a certain syntactic form, and then handling this form with an ad hoc argument. In modern logic texts, Gödel's completeness theorem is usually proved with Henkin 's proof, rather than with Gödel's original proof.
Kurt Gödel (1925) The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a shorter version of the proof, published as an article in 1930, titled "The completeness of the axioms of the functional calculus of logic" (in German)) is not easy to read today; it uses concepts and formalisms that are no longer used and terminology that is often obscure.