A Property Equivalent to the Existence of Scales

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.