Mathematical Logic Quarterly

Subcomplete forcing principles and definable well‐orders

Journal Article

Abstract

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 from MP 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 Δ1‐definable without parameters, and Δ 1 ( H ω 2 ) ‐definable using a subset of ω1 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 of MP SC ( H ω 2 ) mentioned above, and along the way, a bounded forcing axiom for countably closed forcing is proposed.

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