Some consequences of a recursive number-theoretic relation that is not the standard interpretation of any of its formal representations
| dc.creator | Anand, Bhupinder Singh | |
| dc.date | 2002-10-05 | |
| dc.date | 2003-05-11 | |
| dc.date.accessioned | 2026-07-07T04:51:39Z | |
| dc.date.available | 2026-07-07T04:51:39Z | |
| dc.description | We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting inconsistency. We then show: there is an elementary, recursive, number-theoretic relation that is not a formal mathematical object, since it is not the standard interpretation of any of its representations in Goedel's formal system P; the range of a recursive number-theoretic function does not always define a (recursively enumerable set) formal mathematical object consistently in any Axiomatic Set Theory that models P; there is no P-formula, Con(P), whose standard interpretation is unambiguously equivalent to Goedel's number-theoretic definition of "P is consistent"; every recursive number-theoretic function is not strongly representable in P; Tarski's definitions of "satisfiability" and "truth" can be made constructive, and intuitionistically unobjectionable, by reformulating Church's Thesis constructively; the classical definition of Turing machines can be extended to include self-terminating, converging, and oscillating routines; a constructive Church's Thesis implies, firstly, that every partial recursive number-theoretic function can be constructively extended as a unique total function, and, secondly, that we can define effectively computable functions that are not classically Turing-computable; Turing's and Cantor's diagonal arguments do not necessarily define Cauchy sequences. | |
| dc.description | v6; introduced ACI compliant notation for citations; replaced Corollary 1.2; corrected Corollary 14.3; other minor corrections and additions; 76 pages; an HTML version is available at http://alixcomsi.com/CTG_06_Consequences.htm | |
| dc.identifier | https://arxiv.org/abs/math/0210078 | |
| dc.identifier | http://arxiv.org/abs/math/0210078 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/65187 | |
| dc.subject | General Mathematics | |
| dc.subject | 03B10 | |
| dc.title | Some consequences of a recursive number-theoretic relation that is not the standard interpretation of any of its formal representations | |
| dc.type | text |