Mathematical Logic Quarterly

Subcomplete forcing principles and definable well‐orders

Journal Article


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.

Related Topics

Related Publications

Related Content

Site Footer


This website is provided by John Wiley & Sons Limited, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ (Company No: 00641132, VAT No: 376766987)

Published features on are checked for statistical accuracy by a panel from the European Network for Business and Industrial Statistics (ENBIS)   to whom Wiley and express their gratitude. This panel are: Ron Kenett, David Steinberg, Shirley Coleman, Irena Ograjenšek, Fabrizio Ruggeri, Rainer Göb, Philippe Castagliola, Xavier Tort-Martorell, Bart De Ketelaere, Antonio Pievatolo, Martina Vandebroek, Lance Mitchell, Gilbert Saporta, Helmut Waldl and Stelios Psarakis.