FRAGMENTS OF FREGE’SGRUNDGESETZEAND GÖDEL’S CONSTRUCTIBLE UNIVERSE

Abstract

Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem (Theorem 2.9) shows that there is a model of a fragment of the Grundgesetze which defines a model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to Gödel's constructible universe of sets and to Kripke and Platek's idea of the projectum, as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension (Theorem 3.5). As an application, we resolve an analogue of the joint consistency problem in the predicative setting.arXiv:1407.3861v3 [math.LO]

Predicativity, the Russell-Myhill Paradox, and Church’s Intensional Logic

“…In essence, Σ n -separation just says that all the Σ n -subsets of antecedently specified a set exist. Further, it's worth mentioning that the concept of an ordinal α being -admissible from [94] is equivalent to conditions (i)-(ii) of the definition of an intensional position, so that intensional hierarchies are just certain collections of -admissibles for increasing values of . This generalizes the notion of Kripke-Platek set theory since in the case = 1, a structure L α is -admissible just in case it is a model of this set theory ( [56], [69], Devlin [24] p. 48, p. 36).…”

“…Then it can be shown that that M n = L αn for some α with λ < α n < α n+1 < κ and that ρ n (α n ) = λ. For more details on the construction described in this paragraph, see [94], and in particular the existence theorem. 17 17 It's worth spelling out exactly how one defines O n and π n , by more specific reference to the details of the Existence Theorem of [94] and in particular to the function θ n defined therein.…”

“…There are expressive intensional hierarchies (cf. the existence theorem in [94]), and any expressive intensional hierarchy models the Propositions as Fine-Grained as Objects Axiom (3.10). In particular, take the injection χ 1 : α 0 → π −1 1 ({0, 1}).…”

“…This paper is the second in a series of three papers -the other two being [94], [93]-which collectively constitute a sequel to the "Basic Law V" component of our earlier paper [92]. In the companion paper [94], we use Gödel's constructible sets to study how much of Zermelo-Fraenkel set theory can be consistently realized in these fragments of the Grundgesetze. In the complementary paper [93], we examine the deductive strength of a related theory of abstraction principles.…”

“…However, despite these connections to our earlier papers, this paper has been written so that one need not have read these other papers. At one point in §5 below, we reference the earlier paper [94] for examples of one of our defined notions in this paper (namely that of an intensional hierarchy (5.9)). However, this is the only respect in which this paper depends on the earlier papers.…”