Gödel's Incompleteness Theorem by V. A. Uspensky

By V. A. Uspensky

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Additional info for Gödel's Incompleteness Theorem

Sample text

A completely satisfactory execution of this task would require us to make the notion of a proof much more concrete. This would take us beyond the scope of this short book. Nevertheless, in this appendix we shall take a few steps in this direction. 2. Syntactic consistency and syntactic completeness Suppose that (P, P, S) is a deductive system over the alphabet A of the language of arithmetic. ) We shall say that the deductive system is syntactically consistent if there does not exist a closed formula CYfor which both z and 1~1 are provable in the deductive system.

Suppose that c( is a formula in the extended language of arithmetic having numerical parameters xP1, . . , xp,, one-place functional parameters vq,, . , vq,, and twoplace functional parameters w,, , . . , w,>. Let /3 be a formula of the (usual) language are a subset V,‘,, . . , of arithmetic whose parameters of {xP,, . . A. Uspensky Vii, . , I’in, . . ,, . . , W:,, . , W;, . , Wr”,}. Then /I is called a translation of c( if, for any natural numbers xP1, . . ,xp,, V,i, . . , I$,,,, Wr’,, .

Any everywhere defined extension could not be computable. A. Uspensk? We now state a criterion notion of inseparability. l. formulas in a given deductive system to be uncompletable closed formulas which uses the and the set of refutable system are inseparable, then this deductive closed system is uncompletable. Proof. If the deductive system had a completion, then the set of provable closed formulas and the set of refutable closed formulas in the completion would be disjoint enumerable sets which together exhaust the set of all closed formulas.

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