Internal and external consistency of arithmetic

Logica Trianguli 5:19-41 (2001)
Yvon Gauthier
Université de Montréal
What Gödel referred to as “outer” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of arithmetization points rather to an “internalisation” of consistency. The programme was continued by Herbrand, Gödel and Tarski. Tarski’s method of quantifier elimination and Gödel’s Dialectica interpretation are part and parcel of Hilbert’s finitist ideal which is achieved by going back to Kronecker’s programme of a general arithmetic of forms or homogeneous polynomials. The paper can be seen as a historical complement to our result on “The Internal Consistency of Arithmetic with Infinite Descent” . An internal consistency proof for arithmetic means that transfinite induction is not needed and that arithmetic can be shown to be consistent within the bounds of arithmetic, that is with the help of Fermat’s infinite descent and Kronecker’s general or polynomial arithmetic, thus returning into arithmetic without the detour of Cantor’s transfinite arithmetic of ideal elements
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