Much mathematical writing exists that is, explicitly or implicitly, based on set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In ZF, the domain of discourse contains only sets, and hence every mathematical object must be a set. Consequently, in ZF with the usual encoding of an ordered pair a, b , formulas like {a} ∈ a, b have truth values, and operations like P( a, b ) have results that are sets. Such 'accidental theorems' do not match how people think about the mathematics and also cause practical difficulties when using set theory in machine-assisted theorem proving. In contrast, in a number of proof assistants, mathematical objects and concepts can be built of type-theoretic stuff so that many mathematical objects can be, in essence, terms of an extended typed λ-calculus. However, dilemmas and frustration arise when formalizing mathematics in type theory.Motivated by problems of formalizing mathematics with (1) purely set-theoretic and (2) type-theoretic approaches, we explore an option with much of the flexibility of set theory and some of the useful features of type theory. We present ZFP: a modification of ZF that has ordered pairs as primitive, non-set objects. ZFP has a more natural and abstract axiomatic definition of ordered pairs free of any notion of representation. This paper presents axioms for ZFP, and a proof in ZF (machine-checked in Isabelle/ZF) of the existence of a model for ZFP, which implies that ZFP is consistent if ZF is. We discuss the approach used to add this abstraction barrier to ZF.
A generalized set theory (GST) is like a standard set theory but also can have non-set structured objects that can contain other structured objects including sets. This paper presents Isabelle/HOL support for GSTs, which are treated as type classes that combine features that specify kinds of mathematical objects, e.g., sets, ordinal numbers, functions, etc. GSTs can have an exception feature that eases representing partial functions and undefinedness. When assembling a GST, extra axioms are generated following a user-modifiable policy to fill specification gaps. Specialized type-like predicates called soft types are used extensively. Although a GST can be used without a model, for confidence in its consistency we build a model for each GST from components that specify each feature's contribution to each tier of a von-Neumann-style cumulative hierarchy defined via ordinal recursion, and we then connect the model to a separate type which the GST occupies.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.