By the time Hilbert's first series of published research on the foundations of mathematics came to an end around 1904, he had formulated but not solved the problem of finding — for a formal system of arithmetic such as that supplied by Peano — proof of consistency not reliant on the construction of a model. Even if he succeeded in obtaining such a non-semantic proof, there would remain two substantial objections to placing any philosophical significance on the result. First is Frege's objection that the consistency of a list of axioms does not in itself ensure the existence of anything satisfying them. Second is Poincaré's objection that the proof of consistency must make use of the principle of mathematical induction, and is therefore inherently circular. This chapter studies how Hilbert maintained the centrality to his project of finding a consistency proof for arithmetic while adopting a sort of formalism that transformed the significance of such a proof so as to address these two objections.
Keywords: real arithmetic, schematic arithmetic, ideal arithmetic, mathematics, Peano, formal consistency