Hilbert and the internal logic of mathematics

Synthese 101 (1):1 - 14 (1994)
Abstract
Hilbert's programme is shown to have been inspired in part by what we can call Kronecker's programme in the foundations of an arithmetic theory of algebraic quantities.While finitism stays within the bounds of intuitive finite arithmetic, metamathematics goes beyond in the hope of recovering classical logic. The leap into the transfinite proved to be hazardous, not only from the perspective of Gödel's results, but also from a Kroneckerian point of view.
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