 In the time following Frege's revolution, there appears to have been a widespread belief that the set of logical truths of any Fregean language could be characterized as the set of formulae derivable in some suitably chosen calculus (hence, essentially, as the set of numbers obtainable by certain arithmetical operations). Frege himself says, speaking of the higher-order language in his “Begriffsschrift”, that through formalization (in the third sense above) “we arrive at a small number of laws in which, if we add those contained in the rules, the content of all the laws is included, albeit in an undeveloped state” (Frege 1879, §13). The idea follows straightforwardly from Russell's conception of mathematics and logic as identical (see Russell 1903, ch. I, §10; Russell 1920, pp. 194–5) and his thesis that “by the help of ten principles of deduction and ten other premises of a general logical nature (…), all mathematics can be strictly and formally deduced” (Russell 1903, ch. I, §4). See also Bernays (1930, p. 239): “[through formalization] it becomes evident that all logical inference (…) can be reduced to a limited number of logical elementary processes that can be exactly and completely enumerated”. In the opening paragraphs of his paper on logical consequence, Tarski (1936a, 1936b) says that the belief was prevalent before the appearance of Gödel's incompleteness theorems (see subsection 2.4.3 below for the bearing of these theorems on this issue). In recent times, apparently due to the influence of Tarskian arguments such as the one mentioned towards the end of subsection 2.4.3, the belief in the adequacy of derivability characterizations seems to have waned (see e.g. Prawitz 1985 for a similar appraisal).