Abstract
Abstract. We study principles of the form: if a name sigma is forced to have a
certain property phi, then there is a ground model filter g such that sigma^g satisfies phi. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are:
• Any forcing axiom can be expressed as a name principle. For instance,
PFA is equivalent to:
x A principle for rank 1 names (equivalently, nice names) for subsets
of !1.
x A principle for rank 2 names for sets of reals.
• lambdabounded forcing axioms are equivalent to name principles. Bagaria’s
characterisation of BFA via generic absoluteness is a corollary.
We further systematically study name principles where phi is a notion of
largeness for subsets of omega_1 (such as being unbounded, stationary or in the
club filter) and corresponding forcing axioms.
certain property phi, then there is a ground model filter g such that sigma^g satisfies phi. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are:
• Any forcing axiom can be expressed as a name principle. For instance,
PFA is equivalent to:
x A principle for rank 1 names (equivalently, nice names) for subsets
of !1.
x A principle for rank 2 names for sets of reals.
• lambdabounded forcing axioms are equivalent to name principles. Bagaria’s
characterisation of BFA via generic absoluteness is a corollary.
We further systematically study name principles where phi is a notion of
largeness for subsets of omega_1 (such as being unbounded, stationary or in the
club filter) and corresponding forcing axioms.
Original language  English 

Journal  Annals of Pure and Applied Logic 
Publication status  Accepted/In press  12 Dec 2022 
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Dive into the research topics of 'Forcing axioms via ground model interpretations '. Together they form a unique fingerprint.Student theses

Name principles, and hierarchies of regular cardinals applied to LST numbers and inner model theory
Author: Turner, C. J., 24 Jan 2023Supervisor: Welch, P. D. (Supervisor)
Student thesis: Doctoral Thesis › Doctor of Philosophy (PhD)
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