Combining Type Theory and Untyped Set Theory

2006 ◽  
pp. 205-219 ◽  
Author(s):  
Chad E. Brown
Keyword(s):  
Set Theory ◽  
Type Theory

scholarly journals Homotopy type-theoretic interpretations of constructive set theories

2019 ◽  
pp. 1-32
Author(s):  
Cesare Gallozzi
Keyword(s):  
Comparative Analysis ◽  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Homotopy Type Theory ◽  
Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

scholarly journals Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 432 ◽  
Author(s):  
Vilém Novák
Keyword(s):  
Fuzzy Logic ◽  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Rough Sets ◽  
Type Theory ◽  
Rough Set Theory ◽  
Basic Concepts ◽  
Alternative Set

In this paper, we will visit Rough Set Theory and the Alternative Set Theory (AST) and elaborate a few selected concepts of them using the means of higher-order fuzzy logic (this is usually called Fuzzy Type Theory). We will show that the basic notions of rough set theory have already been included in AST. Using fuzzy type theory, we generalize basic concepts of rough set theory and the topological concepts of AST to become the concepts of the fuzzy set theory. We will give mostly syntactic proofs of the main properties and relations among all the considered concepts, thus showing that they are universally valid.

Extending Gödel's negative interpretation to ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 221-229 ◽  
Author(s):  
William C. Powell
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Stable Sets ◽  
Class Theory ◽  
Heyting Arithmetic ◽  
Inner Model ◽  
Order Arithmetic ◽  
Definition Of ◽  
Excluded Middle ◽  
Negative Sense

In [5] Gödel interpreted Peano arithmetic in Heyting arithmetic. In [8, p. 153], and [7, p. 344, (iii)], Kreisel observed that Gödel's interpretation extended to second order arithmetic. In [11] (see [4, p. 92] for a correction) and [10] Myhill extended the interpretation to type theory. We will show that Gödel's negative interpretation can be extended to Zermelo-Fraenkel set theory. We consider a set theory T formulated in the minimal predicate calculus, which in the presence of the full law of excluded middle is the same as the classical theory of Zermelo and Fraenkel. Then, following Myhill, we define an inner model S in which the axioms of Zermelo-Fraenkel set theory are true.More generally we show that any class X that is (i) transitive in the negative sense, ∀x ∈ X∀y ∈ x ¬ ¬ x ∈ X, (ii) contained in the class St = {x: ∀u(¬ ¬ u ∈ x→ u ∈ x)} of stable sets, and (iii) closed in the sense that ∀x(x ⊆ X ∼ ∼ x ∈ X), is a standard model of Zermelo-Fraenkel set theory. The class S is simply the ⊆-least such class, and, hence, could be defined by S = ⋂{X: ∀x(x ⊆ ∼ ∼ X→ ∼ ∼ x ∈ X)}. However, since we can only conservatively extend T to a class theory with Δ01-comprehension, but not with Δ11-comprehension, we will give a Δ01-definition of S within T.

scholarly journals W-types in homotopy type theory

2014 ◽  
Vol 25 (5) ◽  
pp. 1100-1115 ◽  
Author(s):  
BENNO VAN DEN BERG ◽  
IEKE MOERDIJK
Keyword(s):  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Detailed Account ◽  
Simplicial Sets ◽  
Homotopy Type Theory ◽  
Simplicial Presheaves ◽  
Models Of Set Theory

We will give a detailed account of why the simplicial sets model of the univalence axiom due to Voevodsky also models W-types. In addition, we will discuss W-types in categories of simplicial presheaves and an application to models of set theory.

Stewart Shapiro. Introduction—intensional mathematics and constructive mathematics. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, vol. 113, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 1–10. - Stewart Shapiro. Epistemic and intuitionistic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 11–46. - John Myhill. Intensional set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 47–61. - Nicolas D. Goodman. A genuinely intensional set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 63–79. - Andrej Ščedrov. Extending Godel's modal interpretation to type theory and set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 81–119. - Robert C. Flagg. Church's thesis is consistent with epistemic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 121–172.

1991 ◽  
Vol 56 (4) ◽  
pp. 1496-1499 ◽  
Author(s):  
Craig A. Smoryński
Keyword(s):  
New York ◽  
Set Theory ◽  
Type Theory ◽  
Foundations Of Mathematics ◽  
Constructive Mathematics ◽  
Modal Interpretation ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Epistemic Arithmetic ◽  
Intuitionistic Arithmetic

scholarly journals CROSS-WORLD PREDICATION: TYPE-THEORETICAL AND SET-THEORETICAL FORMALIZATION

2021 ◽  
pp. 273-282
Author(s):  
Олег Анатольевич Доманов
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Dependent Types ◽  
Potential Problems ◽  
Traditional Approaches

В статье описываются некоторые потенциальные проблемы теории кросс-мировой предикации Е. Борисова и предлагаются альтернативные формализации в терминах теории типов с зависимыми типами и теории множеств. Преимущество теоретико-типовой формализации состоит в её простоте, связанной с наличием в теории типов функций в зависимые типы. Преимущество предлагаемой теоретико-множественной формализации состоит в большей близости к традиционным подходам и отсутствии некоторых неинтуитивных следствий, таких как предикация по несуществующим объектам. The paper examines some potential problems of the theory of cross-world predication by E. Borisov and suggests alternative formalizations in terms of type theory with dependent types and set theory. The advantage of the type-theoretical formalization lies in its simplicity based on the presence of functions to dependent types. The advantage of the proposed set-theoretical formalization is a greater closeness to traditional approaches and the lack of some non-intuitive effects such as the predication on non-existing objects.

scholarly journals Homotopy Type Theory: A Synthetic Approach to Higher Equalities

2018 ◽  
Author(s):  
Michael Shulman
Keyword(s):  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Synthetic Approach ◽  
Homotopy Type Theory ◽  
Inductive Types ◽  
Basic Ideas ◽  
Univalent Foundations ◽  
Modern Mathematics

Homotopy type theory and univalent foundations (HoTT/UF) is a new foundation of mathematics, based not on set theory but on “infinity-groupoids”, which consist of collections of objects, ways in which two objects can be equal, ways in which those ways-to-be-equal can be equal, ad infinitum. Though apparently complicated, such structures are increasingly important in mathematics. Philosophically, they are an inevitable result of the notion that whenever we form a collection of things, we must simultaneously consider when two of those things are the same. The “synthetic” nature of HoTT/UF enables a much simpler description of infinity groupoids than is available in set theory, thereby aligning with modern mathematics while placing “equality” back in the foundations of logic. This chapter will introduce the basic ideas of HoTT/UF for a philosophical audience, including Voevodsky’s univalence axiom and higher inductive types.

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