These two reasons have at least five immediate consequences for Wittgenstein’s Philosophy of Mathematics.
Rejection of Infinite Mathematical Extensions: Given that a mathematical extension is a symbol (‘sign’) or a finite concatenation of symbols extended in space, there is a categorical difference between mathematical intensions and (finite) mathematical extensions, from which it follows that “the mathematical infinite” resides only in recursive rules (i.e., intensions). An infinite mathematical extension (i.e., a completed, infinite mathematical extension) is a contradiction-in-terms
Rejection of Unbounded Quantification in Mathematics: Given that the mathematical infinite can only be a recursive rule, and given that a mathematical proposition must have sense, it follows that there cannot be an infinite mathematical proposition (i.e., an infinite logical product or an infinite logical sum).
Algorithmic Decidability vs. Undecidability: If mathematical extensions of all kinds are necessarily finite, then, in principle, all mathematical propositions are algorithmically decidable, from which it follows that an “undecidable mathematical proposition” is a contradiction-in-terms. Moreover, since mathematics is essentially what we have and what we know, Wittgenstein restricts algorithmic decidability to knowing how to decide a proposition with a known decision procedure.
Anti-Foundationalist Account of Real Numbers: Since there are no infinite mathematical extensions, irrational numbers are rules, not extensions. Given that an infinite set is a recursive rule (or an induction) and no such rule can generate all of the things mathematicians call (or want to call) “real numbers”, it follows that there is no set of ‘all’ the real numbers and no such thing as the mathematical continuum.
Rejection of Different Infinite Cardinalities: Given the non-existence of infinite mathematical extensions, Wittgenstein rejects the standard interpretation of Cantor’s diagonal proof as a proof of infinite sets of greater and lesser cardinalities.
Since we invent mathematics in its entirety, we do not discover pre-existing mathematical objects or facts or that mathematical objects have certain properties, for “one cannot discover any connection between parts of mathematics or logic that was already there without one knowing” (PG 481). In examining mathematics as a purely human invention, Wittgenstein tries to determine what exactly we have invented and why exactly, in his opinion, we erroneously think that there are infinite mathematical extensions.
If, first, we examine what we have invented, we see that we have invented formal calculi consisting of finite extensions and intensional rules. If, more importantly, we endeavour to determine why we believe that infinite mathematical extensions exist (e.g., why we believe that the actual infinite is intrinsic to mathematics), we find that we conflate mathematical intensions and mathematical extensions, erroneously thinking that there is “a dualism” of “the law and the infinite series obeying it” (PR §180). For instance, we think that because a real number “endlessly yields the places of a decimal fraction” (PR §186), it is “a totality” (WVC 81–82, note 1), when, in reality, “[a]n irrational number isn’t the extension of an infinite decimal fraction,… it’s a law” (PR §181) which “yields extensions” (PR §186). A law and a list are fundamentally different; neither can ‘give’ what the other gives (WVC 102–103). Indeed, “the mistake in the set-theoretical approach consists time and again in treating laws and enumerations (lists) as essentially the same kind of thing” (PG 461).
Closely related with this conflation of intensions and extensions is the fact that we mistakenly act as if the word ‘infinite’ is a “number word”, because in ordinary discourse we answer the question “how many?” with both (PG 463; cf. PR §142). But “‘[i]nfinite’ is not a quantity”, Wittgenstein insists (WVC 228); the word ‘infinite’ and a number word like ‘five’ do not have the same syntax. The words ‘finite’ and ‘infinite’ do not function as adjectives on the words ‘class’ or ‘set’, (WVC 102), for the terms “finite class” and “infinite class” use ‘class’ in completely different ways (WVC 228). An infinite class is a recursive rule or “an induction”, whereas the symbol for a finite class is a list or extension (PG 461). It is because an induction has much in common with the multiplicity of a finite class that we erroneously call it an infinite class (PR §158).
In sum, because a mathematical extension is necessarily a finite sequence of symbols, an infinite mathematical extension is a contradiction-in-terms. This is the foundation of Wittgenstein’s finitism. Thus, when we say, e.g., that “there are infinitely many even numbers”, we are not saying “there are an infinite number of even numbers” in the same sense as we can say “there are 27 people in this house”; the infinite series of natural numbers is nothing but “the infinite possibility of finite series of numbers”—“[i]t is senseless to speak of the whole infinite number series, as if it, too, were an extension” (PR §144). The infinite is understood rightly when it is understood, not as a quantity, but as an “infinite possibility” (PR §138).
Given Wittgenstein’s rejection of infinite mathematical extensions, he adopts finitistic, constructive views on mathematical quantification, mathematical decidability, the nature of real numbers, and Cantor’s diagonal proof of the existence of infinite sets of greater cardinalities.
Since a mathematical set is a finite extension, we cannot meaningfully quantify over an infinite mathematical domain, simply because there is no such thing as an infinite mathematical domain (i.e., totality, set), and, derivatively, no such things as infinite conjunctions or disjunctions (G.E. Moore 1955: 2–3; cf. AWL 6; and PG 281).