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Logical Undecidabilities Made Easy

Leivan D.

Fundamenta Informaticae

2008

Vol. 89, nr 2-3

307-312

Referring to symbolic computing permits extremely simple proofs of undecidability for first-order logic and theories of basic structures. Using grammars we obtain a trivial proof that firstorder validity is undecidable (already for formulas with quantifier prefix ∃∃ and referring to only one unary relation and one binary function [2]). A similar proof yields Gödel's First Incompleteness Theorem for the structure of strings; undecidability for arithmetic follows by noting that addition and multiplication permit direct coding of string concatenation.