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| Content Provider | Springer Nature Link |
|---|---|
| Author | Jacquette, Dale |
| Copyright Year | 2014 |
| Abstract | 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. |
| Starting Page | 239 |
| Ending Page | 262 |
| Page Count | 24 |
| File Format | |
| ISSN | 09254560 |
| Journal | Journal for General Philosophy of Science |
| Volume Number | 45 |
| Issue Number | 2 |
| e-ISSN | 15728587 |
| Language | English |
| Publisher | Springer Netherlands |
| Publisher Date | 2014-02-26 |
| Publisher Place | Dordrecht |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Cantor, Georg Cardinality Computability Computable sequence Denumerability, nondenumerability Diagonalization Entscheidungsproblem Löwenheim-Skolem theorem Skolem paradox Turing, A. N. Universal Turing machine Philosophy of Science History Philosophy of Education Methodology of the Social Sciences Social Sciences |
| Content Type | Text |
| Resource Type | Article |
| Subject | Philosophy Social Sciences History and Philosophy of Science |
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