Reflection and Recurrence

TL;DR

This paper explores how reflection principles and recurrence axioms extend ZFC set theory, aiming for a maximal consistent framework that determines the continuum's size as leph_2 and encompasses many known set-theoretic statements.

Contribution

It introduces the concept of Laver-generic Maximum (LGM), a natural extension of ZFC combining reflection and recurrence principles to potentially serve as an ultimate set-theoretic universe.

Findings

Maximal consistent combination of principles sets the continuum size to leph_2

LGM integrates known ZFC statements as consequences or theorems in many grounds

Reflection and recurrence principles are promising candidates for extending ZFC

Abstract

We examine the Zermelo Fraenkel set theory with Choice (ZFC) enhanced by one of the (structural) reflection principles down to a small cardinal and/or Recurrence Axioms defined below. The strongest forms of reflection principles spotlight the three scenarios in which the size of the continuum is either $\aleph_1$, or $\aleph_2$, or very large, while the maximal setting of Recurrence Axioms points to the set-theoretic universe with the continuum of size $\aleph_2$. We discuss that both the Reflection Principles and Recurrence Axioms can be construed as preferable candidates of the extension of ZFC in terms of the criteria of G\"odel's Program. From this view point, the maximal possible (consistent) combination of these principles and axioms, or even some natural strengthening of the combination (which we want to call ``Laver-generic Maximum'' (LGM)) may be considered as the ultimate extension of ZFC (of course ``ultimate'' only for now -- because of the Incompleteness Theorems): LGM resolves the size of the continuum to be $\aleph_2$ and integrates practically all known statements consistent with ZFC in itself either as its consequences (as it is the case with Martin's Maximum$^{++}$) or as theorems holding in many grounds of the universe (as it is the case with Cicho\'n's Maximum).