Tractatus Logico-Philosophicus. As described by Russell in the Introduction to the Second Edition of PM: "There is another course, recommended by Wittgenstein† (†Tractatus Logico-Philosophicus, *5.54ff) for philosophical reasons. This is to assume that functions of propositions are always truth-functions, and that a function can only occur in a proposition through its values. [...] [Working through the consequences] it appears that everything in Vol. I remains true (though often new proofs are required); the theory of inductive cardinals and ordinals survives; but it seems that the theory of infinite Dedekindian and well-ordered series largely collapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with. Also Cantor's proof that 2n > n breaks down unless n is finite." (PM 2nd edition reprinted 1962:xiv, also cf. new Appendix C). In other words, the fact that an infinite list cannot realistically be specified means that the concept of "number" in the infinite sense (i.e. the continuum) cannot be described by the new theory proposed in PM Second Edition. Wittgenstein in his Lectures on the Foundations of Mathematics, Cambridge 1939 criticised Principia on various grounds, such as: It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and Principia, this would be treated as evidence of an error in Principia (e.g., that Principia did not characterise numbers or addition correctly), not as evidence of an error in everyday counting. The calculating methods in Principia can only be used in practice with very small numbers. To calculate using large numbers (e.g., billions), the formulae would become too long, and some short-cut method would have to be used, which would no doubt rely on everyday techniques such as counting (or else on non-fundamental and hence questionable methods such as induction). So again Principia depends on everyday techniques, not vice versa. Wittgenstein did, however, concede that Principia may nonetheless make some aspects of everyday arithmetic clearer. Gödel 1944 In his 1944 Russell's mathematical logic, Gödel offers a "critical but sympathetic discussion of the logicistic order of ideas": "It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations (contained in *1-*21 of Principia) that it represents in this respect a considerable