Acyclic Comprehension is equal to Stratified Comprehension

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