Complete Totalities

TL;DR

This paper proposes a novel approach to set theory called 'concurrent aggregation' that embraces circularities and totalities, challenging traditional restrictions and offering new insights into set formation and paradoxes.

Contribution

It introduces a new framework for understanding sets as totalities through concurrent aggregation, addressing limitations of traditional set theories and the role of circularities.

Findings

Highlights the limitations of hierarchical set conceptions.

Proposes a new set-theoretic approach embracing circularities.

Provides a foundation for analyzing set paradoxes and totalities.

Abstract

The cumulative hierarchy conception of set, which is based on the conception that sets are inductively generated from "former" sets, is generally considered a good way to create a set conception that seems safe from contradictions. This imposes two restrictions on sets. One is a "limitation of size," and the other is the rejection of non-well-founded sets. Quine's NF system of axioms, does not have any of the two restrictions, but it has a formal restriction on allowed formulas in its comprehension axiom schema, which reflects a similar notion of elements being prior to sets. Here we suggest that a possible reason for set antinomies is the tension between our perception of sets as entities formed from elements by an imaginary aggregation operator, and our wish to regard sets as existing "at once." A new approach to sets as totalities is presented based on a notion of "concurrent aggregation," which instead of avoiding "viscous circles," acknowledges the inherent circularities of some predicates, and provides a way to characterize and investigate these circularities.