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Transactions of the American Mathematical Society

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A property equivalent to the existence of scales


Author: Howard Becker
Journal: Trans. Amer. Math. Soc. 287 (1985), 591-612
MSC: Primary 03E60; Secondary 03E15
DOI: https://doi.org/10.1090/S0002-9947-1985-0768727-6
MathSciNet review: 768727
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Abstract: Let $ {\text{UNIF}}$ and $ {\text{SCALES}}$ be the propositions that every relation on $ {\mathbf{R}}$ can be uniformized, and every subset of $ {\mathbf{R}}$ admits a scale, respectively. For $ A \subset {\mathbf{R}}$, let $ w(A)$ denote the Wadge ordinal of $ A$, and let $ \delta _1^1(A)$ be the supremum of the ordinals realized in the pointclass $ {\Delta ^1}_1(A)$.

Theorem $ {\text{(AD)}}$. The following are equivalent:

(a) $ {\text{SCALES}}$,

(b) $ {\text{UNIF}} + $ the set $ \{ w(A):\delta _1^1(A) = {(w(A))^ + }\} $ contains an $ \omega $-cub subset of $ \Theta $.

Using this theorem, Woodin has shown that if the theory $ {\text{(ZF}} + {\text{DC}} + {\text{AD}} + {\text{UNIF)}}$ is consistent, then the theory $ {\text{(ZF}} + {\text{DC}} + {\text{AD}}_{\mathbf{R}} + {\text{SCALES)}}$ is also consistent. In this paper we give a proof of the above theorem and of a local version of it. We also study the ordinal $ \delta _1^1(A)$ and give several characterizations of it.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0768727-6
Article copyright: © Copyright 1985 American Mathematical Society

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