Results 1 
2 of
2
The Unknowable
, 1999
"... In the early twentieth century two extremely influential research programs aimed to establish solid foundations for mathematics with the help of new formal logic. The logicism of Gottlob Frege and Bertrand Russell claimed that all mathematics can be shown to be reducible to logic. David Hilbert and ..."
Abstract

Cited by 43 (2 self)
 Add to MetaCart
In the early twentieth century two extremely influential research programs aimed to establish solid foundations for mathematics with the help of new formal logic. The logicism of Gottlob Frege and Bertrand Russell claimed that all mathematics can be shown to be reducible to logic. David Hilbert and his school in turn intended to demonstrate, using logical formalization, that the use of infinistic, settheoretical methods in mathematics—viewed with suspicion by many—can never lead to finitistically meaningful but false statements and is thus safe. This came to be known as Hilbert’s program. These grand aims were shown to be impossible by applying the exact methods of logic to itself: the limitative results of Kurt Gödel, Alonzo Church, and Alan Turing in the 1930s revolutionized the whole understanding of logic and mathematics (the key papers are reprinted in [5]). Panu Raatikainen is a fellow in the Helsinki Collegium for Advanced Study and a docent of theoretical philosophy at the University of Helsinki. His email address is
ON INTERPRETING CHAITIN’S INCOMPLETENESS THEOREM
, 1998
"... The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental.