Subcomplete forcing principles and definable well‐orders

It is shown that the boldface maximality principle for subcomplete forcing,MP SC ( H ω 2 ) , together with the assumption that the universe has only set many grounds, implies the existence of a well‐ordering of ℘ ( ω 1 ) definable without parameters. The same conclusion follows fromMP SC ( H ω 2 ) , assuming there is no inner model with an inaccessible limit of measurable cardinals. Similarly, the bounded subcomplete forcing axiom, together with the assumption that x # does not exist, for some x ⊆ ω , implies the existence of a well‐ordering of ℘ ( ω 1 ) which is Δ 1 ‐definable without parameters, andΔ 1 ( H ω 2 ) ‐definable using a subset of ω 1 as a parameter. This well‐order is in L ( ℘ ( ω 1 ) ) . Enhanced versions of bounded forcing axioms are introduced that are strong enough to have the implications ofMP SC ( H ω 2 )mentioned above, and along the way, a bounded forcing axiom for countably closed forcing is proposed.