A Brief History of Future Set Theory

TL;DR

This paper explores the historical and philosophical foundations of naive set theory, proposing that its consistent inference system relies on predicates that produce closed hereditary extensions, and suggests that predicate use with parameters can be understood through meta-mathematical reasoning.

Contribution

It offers a new perspective on naive set theory's consistency by analyzing predicate extensions and introduces a meta-mathematical approach to handling predicates with parameters.

Findings

Naive set theory remains consistent due to predicates forming closed hereditary extensions.

Predicates with free parameters can be effectively managed through meta-mathematical considerations.

The approach supports the use of comprehension schemes without contradiction.

Abstract

Mathematicians still use Naive Set Theory when generating sets without danger of producing any contradiction. Therefore their working method can be considered as a consistent inference system with an experience of over 100 years. My conjecture is that this method works well because mathematicians use only those predicates to form sets, which yield closed hereditary consistent predicate extensions. And for every open formula they use in the process of constructing of a certain (special) set (bottom up), we can always find an "almost-closed" formula (i.e. a parameter-free formula with only the free variable "x") which yields the same certain (special) set as predicate extension as constructed in the bottom up process before. Therefore the use of predicates with free parameters in the Comprehension Scheme does not cause any difficulties and can be "lifted" by meta-mathematical considerations. KEYWORDS: naive set theory, Quine, new foundation, NF, universal sets, comprehension schema, predicate extension, philosophy of set theory, Zermelo, Fraenkel, ZF, complement. CT is the Class-Theoretical frame: logic + "=" + Church Schema + Extensionality Axiom.