The grounded Martin's axiom

TL;DR

The paper introduces the grounded Martin's axiom, a new set-theoretic principle asserting the universe is a ccc forcing extension where Martin's axiom holds for ground model posets, with implications for combinatorics and consistency results.

Contribution

It defines the grounded Martin's axiom, explores its consistency with failures of Martin's axiom, and studies its preservation under forcing extensions, including Cohen and random reals.

Findings

Grounded Martin's axiom is consistent with the failure of Martin's axiom.

Adding a Cohen real preserves the grounded Martin's axiom.

Adding a random real preserves the grounded Martin's axiom even if Martin's axiom fails.

Abstract

We introduce a variant of Martin's axiom, called the grounded Martin's axiom, which asserts that the universe is a ccc forcing extension in which Martin's axiom holds for posets in the ground model. This principle already implies several of the combinatorial consequences of Martin's axiom. The new axiom is shown to be consistent with the failure of Martin's axiom and a singular continuum. We prove that the grounded Martin's axiom is preserved in a strong way when adding a Cohen real and that adding a random real to a model of Martin's axiom preserves the grounded version (even though it destroys Martin's axiom itself). We also consider the analogous variant of the proper forcing axiom.