On The Question Of Absolute Undecidability Essay

2035 WordsNov 9, 20119 Pages
On the Question of Absolute Undecidability⇤ Peter Koellner The incompleteness theorems show that for every su ciently strong consistent formal system of mathematics there are mathematical statements undecided relative to the system.1 A natural and intriguing question is whether there are mathematical statements that are in some sense absolutely undecidable, that is, undecidable relative to any set of axioms that are justified. G¨del was quick to point out that his original incompleteness theorems did o not produce instances of absolute undecidability and hence did not undermine Hilbert’s conviction that for every precisely formulated mathematical question there is a definite and discoverable answer. However, in his subsequent work in set theory, G¨del uncovered what he initially regarded as a o plausible candidate for an absolutely undecidable statement. Furthermore, he expressed the hope that one might actually prove this. Eventually he came to reject this view and, moving to the other extreme, expressed the I am indebted to John Steel and Hugh Woodin for introducing me to the subject and sharing their insights into G¨del’s program. I am also indebted to Charles Parsons o for his work on G¨del, in particular, his 1995. I would like to thank Andr´s Caicedo o e and Penelope Maddy for extensive and very helpful comments and suggestions. I would like to thank Iris Einheuser, Matt Foreman, Haim Gaifman, Kai Hauser, Aki Kanamori, Richard Ketchersid, Paul Larson, and Richard Tieszen, for discussion of these topics. I would also like to thank two referees and Robert Thomas for helpful comments. [Note added June 14, 2009: For this reprinting I have updated the references and added a postscript on recent developments. The main text has been left unchanged apart from the substitution of the Strong ⌦ Conjecture for the ⌦ Conjecture in the statements of certain theorems of

More about On The Question Of Absolute Undecidability Essay

Open Document