|
In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that the arithmetic is consistent – free of any internal contradictions. Hilbert did not state exactly which axioms he considered for arithmetic, except that it was supposed to include the theory of real numbers (Kreisel, 1976). In the 1930s, Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem. Some feel that Gödel's theorems give a negative solution to the problem, while others consider Gentzen's proof as a partial positive solution. == Hilbert's problem and its interpretation == In one English translation, Hilbert asks:
As a nowadays common interpretation, a positive solution to Hilbert's second question would in particular provide a proof that Peano arithmetic is consistent. There are many known proofs that Peano arithmetic is consistent that can be carried out in strong systems such as Zermelo-Fraenkel set theory. These do not provide a resolution to Hilbert's second question, however, because someone who doubts the consistency of Peano arithmetic is unlikely to accept the axioms of set theory (which is much stronger) to prove its consistency. Thus a satisfactory answer to Hilbert's problem must be carried out using principles that would be acceptable to someone who does not already believe PA is consistent. Such principles are often called finitistic because they are completely constructive and do not presuppose a completed infinity of natural numbers. Gödel's incompleteness theorem places a severe limit on how weak a finitistic system can be while still proving the consistency of Peano arithmetic. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hilbert's second problem」の詳細全文を読む スポンサード リンク
|