On universes in type theory

1998 ◽  
Author(s):  
Erik Palmgren
Keyword(s):  
Type Theory ◽  
Second Order ◽  
Separation Axiom ◽  
Second Order Arithmetic ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
Early Results ◽  
Conservation Result ◽  
Constructive Set Theory ◽  
Order Arithmetic

The notion of a universe of types was introduced into constructive type theory by Martin-Löf (1975). According to the propositions-as-types principle inherent in type theory, the notion plays two roles. The first is as a collection of sets or types closed under certain type constructions. The second is as a set of constructively given infinitary formulas. In this paper we discuss the notion of universe in type theory and suggest and study some useful extensions. We assume familiarity with type theory as presented in, for example, Martin-Löf (1984). Universes have been effective in expanding the realm of constructivism. One example is constructive category theory where type universes take the roles of Grothendieck universes of sets, in handling large categories. A more profound example is Aczel’s (1986) type-theoretic interpretation of constructive set theory (CZF). It is done by coding ϵ-diagrams into well-order types, with branching over an arbitrary type of the universe. The latter generality is crucial for interpreting the separation axiom. The introduction of universes and well-orders (W-types) in conjunction gives a great proof-theoretic strength. This has provided constructive justification of strong subsystems of second-order arithmetic studied by proof-theorists (see Griffor and Rathjen (1994) and Setzer (1993), and for some early results, see Palmgren (1992)). At present, it appears that the most easily justifiable way to increase the proof-theoretic strength of type theory is to introduce ever more powerful universe constructions. We will give two such extensions in this paper. Besides contributing to the understanding of subsystems of second-order arithmetic and pushing the limits of inductive definability, such constructions provide intuitionistic analogues of large cardinals (Rathjen et al, in press). A third new use of universes is to facilitate the incorporation of classical reasoning into constructive type theory. We introduce a universe of classical propositions and prove a conservation result for ‘Π-formulas’. Extracting programs from classical proofs is then tractable within type theory. The next section gives an introduction to the notion of universe. The central part of the paper is section 3 where we introduce a universe forming operator and a super universe closed under this operator.

scholarly journals On the existence of Stone-Čech compactification

2010 ◽  
Vol 75 (4) ◽  
pp. 1137-1146 ◽  
Author(s):  
Giovanni Curi
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Natural Extension ◽  
Formal System ◽  
Topological Spaces ◽  
Completely Regular ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
Constructive Set Theory ◽  
In Virtue Of

Introduction. In 1937 E. Čech and M.H. Stone, independently, introduced the maximal compactification of a completely regular topological space, thereafter called Stone-Čech compactification [8, 23]. In the introduction of [8] the non-constructive character of this result is so described: “It must be emphasized that β(S) [the Stone-Čech compactification of S] may be defined only formally (not constructively) since it exists only in virtue of Zermelo's theorem”.By replacing topological spaces with locales, Banaschewski and Mulvey [4, 5, 6], and Johnstone [14] obtained choice-free intuitionistic proofs of Stone-Čech compactification. Although valid in any topos, these localic constructions rely—essentially, as is to be demonstrated—on highly impredicative principles, and thus cannot be considered as constructive in the sense of the main systems for constructive mathematics, such as Martin-Löf's constructive type theory and Aczel's constructive set theory.In [10] I characterized the locales of which the Stone-Čech compactification can be defined in constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural extension of Aczel's system for constructive set theory CZF by a strengthening of the Regular Extension Axiom REA and the principle of Dependent Choice.

scholarly journals Short note: Least fixed points versus least closed points

2021 ◽  
Author(s):  
Gerhard Jäger
Keyword(s):  
Fixed Points ◽  
Short Note ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Arithmetic Operator

AbstractThis short note is on the question whether the intersection of all fixed points of a positive arithmetic operator and the intersection of all its closed points can proved to be equivalent in a weak fragment of second order arithmetic.

UNDECIDABILITY OF THE THEORIES OF CLASSES OF STRUCTURES

2014 ◽  
Vol 79 (4) ◽  
pp. 1001-1019 ◽  
Author(s):  
ASHER M. KACH ◽  
ANTONIO MONTALBÁN
Keyword(s):  
Direct Product ◽  
Disjoint Union ◽  
Second Order ◽  
Linear Orders ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Natural Classes ◽  
Product Of Groups

AbstractMany classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

Formalization of metatheory of the Lambda Calculus in constructive type theory using the Barendregt variable convention

2021 ◽  
pp. 1-20
Author(s):  
Ernesto Copello ◽  
Nora Szasz ◽  
Álvaro Tasistro
Keyword(s):  
Type Theory ◽  
Lambda Calculus ◽  
Proof Assistant ◽  
First Order ◽  
Constructive Type Theory ◽  
Fundamental Part ◽  
Constructive Type ◽  
Free Variables ◽  
The Church ◽  
Theoretical Results

Abstarct We formalize in Constructive Type Theory the Lambda Calculus in its classical first-order syntax, employing only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. As a fundamental part of the formalization, we introduce principles of induction and recursion on terms which provide a framework for reproducing the use of the Barendregt Variable Convention as in pen-and-paper proofs within the rigorous formal setting of a proof assistant. The principles in question are all formally derivable from the simple principle of structural induction/recursion on concrete terms. We work out applications to some fundamental meta-theoretical results, such as the Church–Rosser Theorem and Weak Normalization for the Simply Typed Lambda Calculus. The whole development has been machine checked using the system Agda.

scholarly journals Periodic points and subsystems of second-order arithmetic

1993 ◽  
Vol 62 (1) ◽  
pp. 51-64 ◽  
Author(s):  
Harvey Friedman ◽  
Stephen G. Simpson ◽  
Xiaokang Yu
Keyword(s):  
Periodic Points ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic

Reverse Mathematics: The Playground of Logic

2010 ◽  
Vol 16 (3) ◽  
pp. 378-402 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Mathematical Logic ◽  
Reverse Mathematics ◽  
Computational Approach ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Base Theory ◽  
Order Arithmetic

AbstractThis paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main systems of reverse mathematics. While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above.

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