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:
“In most other modern societies working mothers are not put under these special and exaggerated pressures. For example, French and English mothers often prefer to breast-feed their babies, but they do not feel that their womanhood is at stake if they fail to do so. Nor does anyone else.”
—Sylvia Ann Hewitt (20th century)
“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 (1856–1950)
“If we parents accept that problems are an essential part of life’s challenges, rather than reacting to every problem as if something has gone wrong with universe that’s supposed to be perfect, we can demonstrate serenity and confidence in problem solving for our kids....By telling them that we know they have a problem and we know they can solve it, we can pass on a realistic attitude as well as empower our children with self-confidence and a sense of their own worth.”
—Barbara Coloroso (20th century)