A New Set Theory for Analysis

We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf’s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.

Category theory, introduction to

A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For example, categorical set theory deals with the universe of sets and functions without saying what is in any set, or what any function ‘does to’ anything in its domain; it only talks about the patterns of functions that occur between sets. This stress on patterns of functions originally served to clarify certain working techniques in topology. Grothendieck extended those techniques to number theory, in part by defining a kind of category which could itself represent a space. He called such a category a ‘topos’. It turned out that a topos could also be seen as a category rich enough to do all the usual constructions of set-theoretic mathematics, but that may get very different results from standard set theory.

FRAGMENTS OF FREGE’SGRUNDGESETZEAND GÖDEL’S CONSTRUCTIBLE UNIVERSE

Frege’sGrundgesetzewas 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 theGrundgesetzeformed 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 theGrundgesetze, and our main theorem (Theorem 2.9) shows that there is a model of a fragment of theGrundgesetzewhich 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 theGrundgesetzeare examples ofabstraction 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.

The Problem of Conceptual Understanding

This chapter considers one of the most intriguing questions that philosophy of mathematics in practice must, sooner or later, confront: how understanding of mathematics is obtained. In particular, it examines how issues of meaning and understanding in relation to practice and use relate to the question of the acceptability of “classical” or postulational mathematics, a question usually formulated in terms of consistency. The chapter begins with a discussion of the iterative conception of the universe of sets and its presuppositions, analyzing it from the standpoint of the web of practices. It then addresses the issue of conceptual understanding in mathematics, as exemplifid by the theory Zermelo–Fraenkel axiom system (ZFC). Finally, it looks at arguments based on the idea of the real-number continuum as a source of justification for the axioms of set theory.

Sets in homotopy type theory

Homotopy type theory may be seen as an internal language for the ∞-category of weak ∞-groupoids. Moreover, weak ∞-groupoids model the univalence axiom. Voevodsky proposes this (language for) weak ∞-groupoids as a new foundation for Mathematics called the univalent foundations. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those ‘discrete’ groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of ‘elementary’ of ∞-toposes. We prove that sets in homotopy type theory form a ΠW-pretopos. This is similar to the fact that the 0-truncation of an ∞-topos is a topos. We show that both a subobject classifier and a 0-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover the 0-object classifier for sets is a function between 1-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.

Univalence for inverse diagrams and homotopy canonicity

We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by considering inverse diagrams in Voevodsky's univalent model in simplicial sets, we obtain new models of univalence in a number of (∞, 1)-toposes; this answers a question raised at the Oberwolfach workshop on homotopical type theory. Second, by gluing the syntactic category of univalent type theory along its global sections functor to groupoids, we obtain a partial answer to Voevodsky's homotopy-canonicity conjecture: in 1-truncated type theory with one univalent universe of sets, any closed term of natural number type is homotopic to a numeral.

Incompatible Ω-Complete Theories

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and and are generic extensions of V satisfying CH then and agree on all Σ12-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for Σ12. Moreover, CH is the unique Σ12-statement with this feature in the sense that any other Σ12-statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment Vλ of the universe of sets (for example, one might take Vλ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of Vλ. If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies ¬-CH.