Searcharxiv⌕ Search

arXiv subjects

G. J. Chaitin

Publications and source records attributed to G. J. Chaitin.

30 records · Page 2Linked to original sources
arXiv1995

The Limits of Mathematics---Tutorial Version

The latest in a series of reports presenting the information-theoretic incompleteness theorems of algorithmic information theory via algorithms written in specially designed versions of LISP. Previously in this LISP code only one-character identifiers were allowed, and arithmetic had to be programmed out. Now identifiers can be many characters long, and arithmetic with arbitrarily large unsigned decimal integers is built in. This and many other changes in the software have made this material much easier to understand and to use.

chao-dyn↗
arXiv1994

The Limits of Mathematics---Fourth Version

This is yet another version of the course notes in chao-dyn/9407003. Here we use m-expressions more aggressively to further reduce the constants in our information-theoretic incompleteness theorems. Our main theorems are: 1) an N-bit formal axiomatic system cannot enable one to exhibit any specific object with program-size complexity greater than N+c, 2) an N-bit formal axiomatic system cannot enable one to determine more than N+c' scattered bits of the halting probability Omega. In chao-dyn/9407003, c = 2359 bits and c' = 7581 bits. In chao-dyn/9407005, c = 1127 bits and c' = 3689 bits. In chao-dyn/9407006, c = 994 bits and c' = 3192 bits. And in this version, c = 735 bits and c' = 2933 bits.

chao-dyn↗
arXiv1994

The Limits of Mathematics---Third Version

This is yet another version of the course notes in chao-dyn/9407003. Here we change the universal Turing machine that is used to measure program-size complexity so that the constants in our information-theoretic incompleteness theorems are further reduced. This is done by inventing a more complicated version of lisp in which the parentheses associating defined functions with their arguments can be omitted. This is the third and last version of my course notes. It is not clear to me which is to be preferred, so all three have been made available for comment.

chao-dyn↗
arXiv1994

The Limits of Mathematics---Alternative Version

This is an alternative version of the course notes in chao-dyn/9407003. The previous version is based on measuring the size of lisp s-expressions. This version is based on measuring the size of what I call lisp m-expressions, which are lisp s-expressions with most parentheses omitted. This formulation of algorithmic information theory is harder to understand than the one that was presented in chao-dyn/9407003, but the constants obtained in all theorems are now less than half the size that they were before. It is not clear to me which version of algorithmic information theory is to be preferred.

chao-dyn↗
arXiv1994

The Limits of Mathematics---The Book

This is a revised version of the course notes handed to each participant at the limits of mathematics short course, Orono, Maine, June 1994.

chao-dyn↗
arXiv1994

The Berry Paradox

Lecture given Wednesday 27 October 1993 at a Physics -- Computer Science Colloquium at the University of New Mexico. The lecture was videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa Fe Institute 24--26 May 1994.

chao-dyn↗
arXiv1994

The Limits of Mathematics (in C)

This is a shortened version of "The Limits of Mathematics--Course Outline & Software" (IBM Research Report RC 19324, December 1993) in which all Mathematica code has either been deleted or, if absolutely necessary, replaced by C code. The intention is to make this material available to a wider audience.

chao-dyn↗
arXiv1993

The Limits of Mathematics---Course Outline and Software

A remarkable new definition of a self-delimiting universal Turing machine is presented that is easy to program and runs very quickly. This provides a new foundation for algorithmic information theory. This new universal Turing machine is implemented via software written in Mathematica and C. Using this new software, it is now possible to give a self-contained ``hands on'' mini-course presenting very concretely the latest proofs of the fundamental information-theoretic incompleteness theorems.

chao-dyn↗