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 
Early online date  28 Mar 2023 
Publication status  Epub ahead of print  28 Mar 2023 
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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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