Gödel's theorem may be demonstrated using arguments having an information-theoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.

was Godel's secretary. She said that Godel was very careful about his health and because of the snow he wasn't coming to the Institute that day and therefore my appointment was canceled. And that's how I had two phone conversations with Godel but never met him. I never tried again. I'd like to tell you what I would have told Godel. What I wanted to tell Godel is the difference between what you get when you study the limits of mathematics the way Godel did using the paradox of the liar, and what I get using the Berry paradox instead. What is the paradox of the liar? Well, the paradox of the liar is "This statement is false!" Why is this a paradox? What does "false" mean? Well, "false" means "does not correspond to reality." This statement says that it is false. If that doesn't correspond to reality, it must mean that the statement is true, right? On the other hand, if the statement is true it means that what it says corresponds to reality. But it says that it is false. Therefore the sta

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.

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Effectiveness of Mathematics in the Natural Sciences [Wig60]. This paper can be construed as an examination and affirmation of Galileo’s tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effectiveness of

Using 1947 work of Post showing that the word problem for semigroups is unsolvable, we explicitly exhibit an algebraic characterization of the bits of the halting probability Ω. Our proof closely follows a 1978 formulation of Post’s work by M. Davis. The proof is selfcontained and not very complicated.

We propose using random walks in software space as abstract formal models of biological evolution. The goal is to shed light on biological creativity using toy models of evolution that are simple enough to prove theorems about them. We consider two models: a single mutating piece of software, and a population of mutating software. The fitness function is taken from a well-known problem in computability theory that requires an unlimited amount of creativity, the Busy Beaver problem.

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Efforts to calculate values of the noncomputable Busy Beaver function are discussed in the light of algorithmic information theory. I would like to talk about some impossible problems that arise when one combines information theory with recursive function or computability theory. That is to say, I'd like to look at some unsolvable problems which arise when one examines computation unlimited by any practical 2 G. J. Chaitin bound on running time, from the point of view of information theory. The result is what I like to call "algorithmic information theory" [5]. In the Computer Recreations department of a recent issue of Scientific American [7], A. K. Dewdney discusses efforts to calculate the Busy Beaver function \Sigma. This is a very interesting endeavor for a number of reasons. First of all, the Busy Beaver function is of interest to information theorists, because it measures the capability of computer programs as a function of their size, as a function of the amount of informatio...

2 Chaitin coined the name AIT; this name is becoming more and more popular. vii viii Preface During its history of more than 40 years, AIT knew a significant variation in terminology. In particular, the main measures of complexity studied in AIT were called Solomonoff-Kolmogorov-Chaitin complexity, Kolmogorov-Chaitin complexity, Kolmogorov complexity, Chaitin complexity, algorithmic complexity, program-size complexity, etc. Solovay’s handwritten notes [22] 3, introduced and used the terms Chaitin complexity and Chaitin machine. 4 The book [21] promoted the name Kolmogorov complexity for both AIT and its main complexity. 5 The main contribution shared by AIT founding fathers in the mid 1960s was the new type of complexity—which is invariant up to an additive constant—and, with it, a new way to reason about computation. Founding fathers ’ subsequent contributions varied considerably. Solomonoff’s