Minimum models of second-order set theories

TL;DR

This paper explores the existence and properties of minimal models of various second-order set theories, revealing conditions for their existence and demonstrating the absence of some minimal models.

Contribution

It provides new results on the existence of minimum models for Kelley--Morse, G"odel--Bernays, and related theories, including conditions and non-existence proofs.

Findings

A countable ZFC model has a minimum GBC-realization iff it admits a parametrically definable global well-order.

Countable GBC models admit minimal extensions with the same sets.

No minimum transitive model exists for Kelley--Morse set theory.

Abstract

In this article I investigate the phenomenon of minimum models of second-order set theories, focusing on Kelley--Morse set theory $\mathsf{KM}$, G\"odel--Bernays set theory $\mathsf{GB}$, and $\mathsf{GB}$ augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of $\mathsf{ZFC}$ has a minimum $\mathsf{GBC}$-realization if and only if it admits a parametrically definable global well-order. (2) Countable models of $\mathsf{GBC}$ admit minimal extensions with the same sets. (3) There is no minimum transitive model of $\mathsf{KM}$. (4) There is a minimum $\beta$-model of $\mathsf{GB} + \mathsf{ETR}$. The main question left unanswered by this article is whether there is a minimum transitive model of $\mathsf{GB} + \mathsf{ETR}$.