Computable Diagonalizations and Turing's Cardinality Paradox

A. N. Turing's 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing's Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing's objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing machine, computability, computable sequences, and Turing's effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing's Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing's discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing's two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers

ONTIC: A Knowledge Representation System for Mathematics

Ontic is an interactive system for developing and verifying mathematics. Ontic's verification mechanism is capable of automatically finding and applying information from a library containing hundreds of mathematical facts. Starting with only the axioms of Zermelo-Fraenkel set theory, the Ontic system has been used to build a data base of definitions and lemmas leading to a proof of the Stone representation theorem for Boolean lattices. The Ontic system has been used to explore issues in knowledge representation, automated deduction, and the automatic use of large data bases

A Mechanical Proof of the Turing Completeness of Pure Lisp

We describe a proof by a computer program of the Turing completeness of a computational paradigm akin to Pure LISP. That is, we define formally the notions of a Turing machine and of a version of Pure LISP and prove that anything that can be computed by a Turing machine can be computed by LISP. While this result is straightforward, we believe this is the first instance of a machine proving the Turing completeness of another computational paradigm. The work here was supported in part by NSF Grant MCS-8202943 and ONR Contract N00014-81-K-0634. 2 1. Introduction. In our paper [Boyer & Moore 84] we present a definition of a function EVAL that serves as an interpreter for a language akin to Pure LISP, and we describe a mechanical proof of the unsolvability of the halting problem for this version of LISP. We claim in that paper that we have proved the recursive unsolvability of the halting problem. It has been pointed out by a reviewer that we cannot claim to have mechanically proved the r..