Abstract
By coding Polish metric spaces with metrics on countable sets, we propose an interpretation of Polish metric spaces in models of ZFC and extend Mostowski's classical theorem of absoluteness of analytic sets for any Polish metric space in general. In addition, we prove a general version of Shoenfield's absoluteness theorem.
Highlights
Mostowski’s Absoluteness Theorem states that any analytic subset of the Baire space ωω is absolute for transitive models of ZFC
Shoenfields’s Absoluteness Theorem states that any Σ12 subset of the Baire space is absolute for transitive models M ⊆ N of ZFC when ω1N ⊆ M
The Σ11 absoluteness theorem was proven by Mostowski [9] in 1959, while Σ12 absoluteness was proven by Shoenfield [10] in 1961,1 though proofs with modern notation can be found in standard references like [4] and [8]
Summary
Mostowski’s Absoluteness Theorem ( known as Σ11 absoluteness) states that any analytic subset of the Baire space ωω is absolute for transitive models of ZFC. The same arguments seem to be able to be readapted for spaces like R, Rω, or any other standard Polish spaces It seems that there is no reference of a general version of these absoluteness theorems for arbitrary Polish spaces. It may be that mathematicians trust that they can be reproved for each particular Polish space that comes at hand, so there is no worry to provide general statements Another reason may be that each well known Polish space has a certain shape that tells how to be interpreted in an arbitrary model of ZFC and standard ways of interpreting may vary depending on each space.
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