Abstract. I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein's "form-series" device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named.There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is 1 1 -complete. But third, it is only granted the assumption of countability that the class of tautologies is 1 -definable in set theory.Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects.We have by now a quite systematic and rigorous grasp of the logical work of two of Wittgenstein's teachers, Frege and Russell. This is thanks in part to decades of flourishing scholarship, and thanks also to Frege and Russell themselves. In contrast, despite comparably voluminous commentary there is still no received understanding of anything describable as the logical system of the Tractatus (Wittgenstein, 1921). It is hard to resist the conclusion that the Tractatus did not, despite its professed program and its large reputation, offer any systematic alternative conception of the nature of logic.But the conclusion is mistaken. To the contrary, there is a system, or a class of similar systems, which can be understood to explicate the development of logic in the Tractatus. They differ rather sharply from those of Frege or Russell, as well as from classical first-or second-order logic. Nonetheless they can be exactly described and investigated metamathematically, for epistemological and metaphysical evaluation. In this paper, I will present one such system, and investigate some of its properties. These turn out to be of surprising and indeed independent interest.In seeking to understand what logic is supposed to be according to the early Wittgenstein, we may distinguish two kinds of evidence. First, there are the contours of his own construction, famously, for example, in the truth-functionality thesis and its enactment through iterated joint denial. Second, there are in the Tractatus apparent declarations of epistemological constraints on the nature of logic: some of these, for example, have been taken to suggest that according to Wittgenstein, logic must be decidable.I wish to separate these two strands. In the early decades of the 20th century, it was entirely possible for a proficient researcher to develop a computationally intractable system