Some attempts at a direct reduction of the infinite to the (large) finite

I survey some endeavors which have been made to attain a sort of direct reduction of the usual notion of countable infinity to some reasonable notion of finiteness, in terms of nonstandard arithmetic, feasibility, pseudo-models of derivations, Ehrenfeucht star-models, etc. I maintain that although many interesting results have been obtained in these attempts, they ultimately show that (at least by the means considered here) no satisfactory reduction is possible.