１９ Incompleteness Theorem

１９　Incompleteness Theorem　【What is the Universe？】Dialogue with the Universe

○The proposition which cannot judge correction exists.
・・・Gödel (Kurt Gödel, Austria 1906-78) proved in 1930.
・・・Then, Turing(Aan Mathison Turing, UK 1912-1954) proved that there is no unific method of confirming beforehand whether a certain proposition is a proposition which cannot judge truth.
Reference: 「世にも美しい数学入門」Masahiko Fujiwara／Youko Ogawa(p.147～151)

The first incompleteness theorem
No consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers.

The second incompleteness theorem
An extension of the first, shows that the system cannot demonstrate its own consistency.

☆If there is a proposition which cannot confirm that verification is impossible and it is unverifiable, it can say, "Search is eternal." 
・・・Although nothing may remain even if it continues search throughout life・・・

【Mathematical history】
About 7000 years ago 　Number was born. (Mankind began to group living.)
16th century Imaginary number is accepted.
・・・"Forgetting the mental anguish received with imaginary number, and merely introducing this." Gerolamo Cardano（Italy 1501～1576）
17th century　　Minus is accepted in Europe.
・・・Descartes draws a number line. 　　René Descartes（France 1596～1650）
The first half of the 19th century　　An imaginary number is denoted by an axis of coordinates.：complex plane　　Carolus Fridericus Gauss（Germany 1777～1855）
1935　　Gödel's Incompleteness Theorem
・・・Then, Turing(Aan Mathison Turing, UK 1912-1954) proved that there is no unific method of confirming beforehand whether a certain proposition is a proposition which cannot judge truth.

【References】
1. Gödel's incompleteness theorems - Wikipedia



【Change log】
20170429　Addition of theorems