Abstract

Serious difficulties attend the reading of David Hilbert's 1925 classic paper ?On the infinite?. I claim that the peculiarities of presentation plaguing certain parts of that paper, as well as of the earlier ?On the Foundations of Logic and Arithmetic? (1904), are due to a tension between two incompatible semantical approaches to numerical statements of elementary arithmetic, and accordingly two incompatible metaphysical conceptions of the natural numbers. One of these approaches is the referential, or model-theoretical one; the other is the iterativist's approach. I draw out the two tendencies in these works, with more attention paid to Hilbert's iterativistic tendency because of the unfamiliarity of iterativism generally. I begin with an exposition of this view