On Recurrence Axioms

Abstract

The Recurrence Axiom for a class $P$ of posets and a set A of parameters is an axiom scheme in the language of ZFC asserting that if a statement with parameters from A is forced by a poset in $P$, then there is a ground containing the parameters and satisfying the statement.

The tightly super-$C^{(\infty )}$-$P$-Laver generic hyperhuge continuum implies the Recurrence Axiom for $P$ and $H\left(2^{{\aleph }_0}\right)$. The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly $P$-generic hyperhuge cardinal κ, and that κ in the bedrock is genuinely hyperhuge, or even super $C^{(\infty )}$ hyperhuge if κ is a tightly super-$C^{(\infty )}$-$P$-Laver generic hyperhuge definable cardinal.

The Laver Generic Maximum (LGM), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For instance, double plus version of Martin's Maximum is a consequence of LGM while Cichoń's Maximum is a phenomenon in many grounds of the universe under LGM.