On Interpreting Chaitin's Incompleteness Theorem

Abstract

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.

Ultimate Intelligence Part I: Physical Completeness and Objectivity of Induction

“…An additional nice property of using physical stochastic models, e.g., statistical mechanics, stochastic dynamical systems, quantum computing models, instead of abstract machine or computation models is that we can refute a wellknown objection to algorithmic information by Raatikainen [11], which depends on unnatural enumerations of recursive functions, essentially constructing reference machines with a lot of useless information. Such superfluous reference machines would incur a physical cost in physical message complexity, and therefore they would not be picked by our definition, which is exactly why you cannot shuffle program indices as you like, because such permutations require additional information to encode.…”

Constructivity and Computability in Historical and Philosophical Perspective

“…The previous statement, and Chaitin's assertion that the Kolmogorov complexity of T somehow measures the power of T as a theory, has been much criticized in van Lambalgen (1989), Fallis (1996) and Raatikainen (1998). The previous statement, and Chaitin's assertion that the Kolmogorov complexity of T somehow measures the power of T as a theory, has been much criticized in van Lambalgen (1989), Fallis (1996) and Raatikainen (1998).…”

Kolmogorov complexity and characteristic constants of formal theories of arithmetic

“…Raatikainen [5] focused on the choice of universal Turing machine as a factor which influences the value of c T and he proved that for any recursively axiomatizable consistent theory T there exists a universal Turing machine such that c T is zero; thus c S = c T is always possible for two theories S and T.…”

“…Another characteristic constant r T was focused on by Raatikainen [5] in connection with c T . We show that the difference between c T and r T can be made arbitrarily large by chainging the universal Turing machine, but, on the other hand, that r S < r T for some universal Turing machine if and only if c S < c T for some (possibly different) universal Turing machine.…”