ZF*-Extensionality interprets full ZF

TL;DR

This paper explores whether a version of the Replacement axiom can be formulated to remain valid without Extensionality in ZF set theory, building on Dana Scott's foundational work.

Contribution

It proposes a new form of the Replacement axiom that can withstand the removal of Extensionality in ZF set theory, extending Scott's earlier investigations.

Findings

A new formulation of Replacement compatible with non-Extensional ZF

Demonstrates that Extensionality is crucial for standard Replacement

Provides insights into the foundational structure of set theory

Abstract

Dana Scott had shown that removing Extensionality from ZF set theory formalized in the customary manner would weaken it down to Zermelo set theory. The following proof is my personal attempt to solve the question of whether we can have a version of replacement that can withstand removal of Extensionality when ZF is formalized with it instead of the standard axiom schema of replacement. Dana Scott had investigated that, the general lines of the proof here are similar to his; however, the form of Replacement suggested here is different.