Large Cardinals and Definable Well-Orderings of the Universe

TL;DR

This paper demonstrates how to construct a universe with a definable well-ordering that preserves large cardinal properties and the GCH, using a reverse Easton forcing iteration and coding techniques.

Contribution

It introduces a method to obtain a definable well-ordering of the universe while maintaining large cardinal axioms and the GCH through a novel forcing construction.

Findings

Successfully creates a universe with a definable well-ordering

Preserves proper classes of large cardinals

Maintains the GCH after forcing

Abstract

We use a reverse Easton forcing iteration to obtain a universe with a definable well-ordering, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle diamond star at a proper class of successor cardinals.