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The British Journal for the Philosophy of Science 1953 IV(14):107-129; doi:10.1093/bjps/IV.14.107
© 1953 by British Society for the Philosophy of Science
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ARTICLES

A VARIANT TO HILBERT'S THEORY OF THE FOUNDATIONS OF ARITHMETIC*

G. KREISEL

IN Hilbert's theory of the foundations of any given branch of mathematics the main problem is to establish the consistency (of a suitable formalisation) of this branch. Since the (intuitionist) criticisms of classical logic, which Hilbert's theory was intended to meet, never even alluded to inconsistencies (in classical arithmetic), and since the investigations of Hilbert's school have always established much more than mere consistency, it is natural to formulate another general problem in the foundations of mathematics: to translate statements of theorems and proofs in the branch considered into those of some preferred system, where the translation must satisfy certain appropriate conditions (interpretation). The problem is relative to the choice of preferred system, as is Hilbert's consistency problem since he required the consistency to be established by particular methods (finitist ones).

* Paper read to the Philosophy of Science Group on 9th February 1953


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