 We just noted that the Fregean logician's formalized grammar amounts to an algorithm for producing formulae from the basic artificial symbols. This is meant very literally. As was clear to mathematical logicians from very early on, the basic symbols can be seen as (or codified by) natural numbers, and the formation rules in the artificial grammar can be seen as (or codified by) simple computable arithmetical operations. The grammatical formulae can then be seen as (or codified by) the numbers obtainable from the basic numbers after some finite series of applications of the operations, and thus their set is characterizable in terms of concepts of arithmetic and set theory (in fact arithmetic suffices, with the help of some tricks).