Acyclic Comprehension is equal to Stratified Comprehension

TL;DR

This paper proves that the concept of acyclic comprehension, based on acyclic formulas, is logically equivalent to the well-known stratified comprehension criterion in set theory.

Contribution

The paper establishes the surprising equivalence between acyclic comprehension and stratified comprehension, clarifying their relationship in set theory.

Findings

Acyclic comprehension is equivalent to stratified comprehension.

Acyclic comprehension is implied by stratification.

The equivalence was previously conjectured but not proven.

Abstract

A new criterion of comprehension is defined, initially termed by myself as "connected" and finally as "Acyclic" by Mr. Randall Holmes. Acyclic comprehension simply asserts that for any acyclic formula phi, the set {x:phi} exists. I first presented this criterion semi-formally to Mr. Randall Holmes, who further made the first rigorous definition of it, a definition that I finally simplified to the one presented here. Later Mr. Holmes made another presentation of the definition which is also mentioned here. He pointed to me that acyclic comprehension is implied by stratification, and posed the question as to whether it is equivalent to full stratification or strictly weaker. He and initially I myself thought that it was strictly weaker; Mr. Randall Holmes actually conjectured that it is very weak. Surprisingly it turned to be equivalent to full stratification as I proved here