AD+
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In [[set theory]], '''AD+''' is an extension, proposed by [[W. Hugh Woodin]], to the [[axiom of determinacy]]. The axiom, which is to be understood in the context of [[Zermelo-Fraenkel set theory|ZF]] plus DC<sub>R</sub> (the [[axiom of dependent choice]] for [[real number|reals]]), states two things:
# Every [[set]] of reals is [[∞-Borel]].
# For any ordinal λ less than [[Theta (set theory)|Θ]], any subset ''A'' of ω<sup>ω</sub>, and any [[continuous function]] π:λ<sup>ω</sup>→ω<sup>ω</sup>, the [[preimage]] π<sup>-1</sup><nowiki>[A]</nowiki> is [[Determinacy|determined]]. (Here λ<sup>ω</sup> is to be given the [[product topology]], starting with the [[discrete topology]] on λ.)
The second clause by itself is called '''''ordinal determinacy'''''.
==External links==
*Woodin, W. Hugh (2001). "[http://math.berkeley.edu/~woodin/talks/lc2000.3.pdf The Continuum Hypothesis (III)]". Slide 8. Accessed on [[2 October]], [[2005]].
[[Category:Axioms of set theory]]
[[Category:Determinacy]]
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