Nov 26, 2012

Godel Work to Show the Un-Success of Hilbert's Program_Documented by Marsigit



Godel Work to Show the Un-Success of Hilbert's Program


Gödel showed Hilbert's Program can not succeed. This was proven in what is now called Gödel's Incompleteness Theorem:

Let S be a formal system for number theory. If S is consistent, then there is a sentence, G, such that neither G nor the negation of G (written G) is a theorem of S. Thus, any formal system sufficient to express the theorems of number theory has to be incomplete.

Proof:
S can prove P(n) just in case n is the Gödel-number of a theorem of S. There exists k, such that k is a Gödel-number of the formula P(k)=G. This statement says of itself, it is not provable. Even if we define a new formal system S = S + G (thus including the undecidable theorem as an axiom), we can find G which isn't provable in (is independent of) S. The reasoning Gödel used for his incompleteness theorem is finitary, so it could be formalized inside S. Thus, S can prove that if S is consistent, then G is not provable. Note that the underlined phrase is what G says, so S proves Cst(S)  implies G is true, but G says G is not provable. Suppose S can prove Cst(S), then S can prove G, but if S is consistent, it can't prove G, thus it can't prove its consistency. Thus, Hilbert's Program does not work; one cannot prove the consistency of a mathematical theory.




1 comment:

  1. Syahrial
    16701251015
    S2 PEP kelas B 2016
    Teorema ketaklengkapan Gödel (bahasa Inggris: Gödel's incompleteness theorems) adalah dua teorema logika matematika yang menetapkan batasan (limitation) inheren dari semua kecuali sistem aksiomatik yang paling trivial yang mampu mengerjakan aritmetika. Teorema-teorema ini, dibuktikan oleh Kurt Gödel pada tahun 1931, penting baik dalam logika matematika maupun dalam filsafat matematika. Kedua hasil ini secara luas, tetapi tidak secara universal, ditafsirkan telah menunjukkan bahwa program Hilbert untuk menghitung himpunan lengkap dan konsisten dari aksioma-aksioma bagi semua matematika adalah tidak mungkin, sehingga memberikan jawaban negatif terhadap soal Hilbert yang kedua.

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