We prove the determinacy of all Δ 1 1 games on arbitrary trees, and we use this result and the assumption that a measurable cardinal exists to demonstrate the determinacy of all games on ω ω that belong both to – Π 1 1 and to its dual.

We characterize in terms of determinacy, the existence of the least inner model of “every real has a sharp”. We let #1 be the sharp function on the reals and define two classes of sets, * and *+, which lie strictly between β<ω2- and Δ. We show that the determinacy of * follows from L[#1] “every reak has a sharp”; and we show that the existence of indiscernibles for L[#1] is equivalent to a slightly stronger determinacy hypothesis, the determinacy of (...) *+. (shrink)

This paper introduces the real core model K() and determines the extent of scales in this inner model. K() is an analog of Dodd-Jensen's core model K and contains L(), the smallest inner model of ZF containing the reals R. We define iterable real premice and show that Σ1∩() has the scale property when vR AD. We then prove the following Main Theorem: ZF + AD + V = K() DC. Thus, we obtain the Corollary: If ZF + AD +()L() (...) is consistent, then ZF + AD + DC + α < ω2 -AD is also consistent. (shrink)

For several partial sharp functions # on the reals, we characterize in terms of determinacy, the existence of indiscernibles for several inner models of “# exists for every real r”. Let #10=1#10 be the identity function on the reals. Inductively define the partial sharp function, β#1γ+1, on the reals so that #1γ+1 =1#1γ+1 codes indiscernibles for L [#11, #12,…, #1γ] and #1γ+1=#1γ+1). We sho w that the existence of β#1γ follows from the determinacy of *Σ01)*+ games . Part I proves (...) the converse. (shrink)

DuBose, D.A., Determinacy and the sharp function on the reals, Annals of Pure and Applied Logic 55 237–263. We characterize in terms of determinacy, the existence of the least inner model of “every real has a sharp”. We let 1 be the sharp function on the reals and define two classes of sets, * and *+, which lie strictly between β<ω2 an d Δ. We show that the determinacy of * follows from L[#1] xvR; “every real has a sharp”; and (...) we show that the existence of indiscernibles for L[#1] is equivalent to a slightly determinacy hypothesis, the determinacy of *+. (shrink)