Modern Viewpoints On The Status of The Problem
While the theorems of Gödel and Gentzen are now well understood by the mathematical logic community, no consensus has formed on whether (or in what way) these theorems answer Hilbert's second problem. Simpson (1988:sec. 3) argues that Gödel's incompleteness theorem shows that it is not possible to produce finitistic consistency proofs of strong theories. Kreisel (1976) states that although Gödel's results imply that no finitistic syntactic consistency proof can be obtained, semantic (in particular, second-order) arguments can be used to give convincing consistency proofs. Detlefsen (1990:p. 65) argues that Gödel's theorem does not prevent a consistency proof because its hypotheses might not apply to all the systems in which a consistency proof could be carried out. Dawson (2006:sec. 2) calls the belief that Gödel's theorem eliminates the possibility of a persuasive consistency proof "erroneous", citing the consistency proof given by Gentzen and a later one given by Gödel in 1958.
Read more about this topic: Hilbert's Second Problem
Famous quotes containing the words modern, status and/or problem:
“However far modern science and technics have fallen short of their inherent possibilities, they have taught mankind at least one lesson: Nothing is impossible.”
—Lewis Mumford (18951990)
“The censorship method ... is that of handing the job over to some frail and erring mortal man, and making him omnipotent on the assumption that his official status will make him infallible and omniscient.”
—George Bernard Shaw (18561950)
“The problem of induction is not a problem of demonstration but a problem of defining the difference between valid and invalid
predictions.”
—Nelson Goodman (1906)