Subcomplete forcing principles and definable well‐orders

Mathematical Logic Quarterly 64 (6):487-504 (2018)
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It is shown that the boldface maximality principle for subcomplete forcing,, together with the assumption that the universe has only set many grounds, implies the existence of a well‐ordering of definable without parameters. The same conclusion follows from, assuming there is no inner model with an inaccessible limit of measurable cardinals. Similarly, the bounded subcomplete forcing axiom, together with the assumption that does not exist, for some, implies the existence of a well‐ordering of which is Δ1‐definable without parameters, and ‐definable using a subset of ω1 as a parameter. This well‐order is in. Enhanced versions of bounded forcing axioms are introduced that are strong enough to have the implications of mentioned above, and along the way, a bounded forcing axiom for countably closed forcing is proposed.



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Aronszajn tree preservation and bounded forcing axioms.Gunter Fuchs - 2021 - Journal of Symbolic Logic 86 (1):293-315.

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References found in this work

Set-theoretic geology.Gunter Fuchs, Joel David Hamkins & Jonas Reitz - 2015 - Annals of Pure and Applied Logic 166 (4):464-501.
The downward directed grounds hypothesis and very large cardinals.Toshimichi Usuba - 2017 - Journal of Mathematical Logic 17 (2):1750009.
Set mapping reflection.Justin Tatch Moore - 2005 - Journal of Mathematical Logic 5 (1):87-97.

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