Constructive Universe and Computation
Words and integers: two constructive worlds. (a) In Chapters I and II we have studied alphabets, words (finite sequences of letters of an alphabet), expressions (certain syntactically well formed words such as terms and formulas defined in I.2.3), deductions (finite sequences of formulas defined in II.5.1).
Let us fix an alphabet of a first-order language and denote by W⊃F the sets of words and formulas respectively.
Studying deducibility, we have implicitly introduced the set D⊂F of all formulas deducible from, say, a fixed finite set of formulas (axioms). This whole set D can be systematically generated and well ordered following a finitely describable procedure that, say, first totally orders the alphabet, then totally orders elementary steps of deductions etc., prescribing in what order to apply them iteratively to the axioms and already deduced formulas.
In this way we get a bijection Z+ → D that is intuitively “computable,” together with the inverse bijection. Of course, it is a simple particular case of numbering defined in VII.1.2 and studied later on in VII.1. See also II.11 for a useful numbering of all formulas in the Smullyan language.
Having achieved in this way the encoding of certain linguistic constructions by arithmetic ones, we have been able in Part III to reduce many problems of syntax (and partly semantics) of formal languages to number theory.
KeywordsTuring Machine Recursive Function Kolmogorov Complexity Geometric Realization Quantum Search
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