DOI: 10.14704/nq.2004.2.2.39

Do Gödel’s Incompleteness Theorems Set Absolute Limits on the Ability of the Brain to Express and Communicate Mental Concepts Verifiably?

Bhupinder Singh Anand

Abstract


Classical interpretations of Gödel’s formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is, both, non-algorithmic, and essentially unverifiable. However, a language of general, scientific, discourse, which intends to mathematically express, and unambiguously communicate, intuitive concepts that correspond to scientific investigations, cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Gödel’s reasoning, under which any formal system of Peano Arithmetic - classically accepted as the foundation of all our mathematical languages - is verifiably complete in the above sense. We show how some paradoxical concepts of Quantum mechanics can, then, be expressed, and interpreted, naturally under a constructive definition of mathematical truth.

Keywords


algorithm; arithmetic; Cantor; Cauchy; Church; classical; complete; computable; consistent; constructive; Dedekind; diagonal method; effective method; expressible; formal language; Gödel; halting; individually true; interpretation; intuitive; looping

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