Low-level Determinacy within \Delta(\omega² - \Pi¹₁)
by
Derrick Dubose
University of Nevada

For various sequences [#\vec] of sharp functions there corresponds a class \Gamma([#\vec]) such that:

(*) the existence of indiscernibles for > L[[#\vec]] is equivalent to the determinacy of (\Gamma([#\vec]))^*₊ games.

For instance, it has been known for some time that (*) holds when k < \omega, [#\vec] = #^k, and \Gamma(#^k) = _df \Pi⁰_k, where #^k is the sharp function on objects of type k. More generally, such classes \Gamma([#\vec]) are known for sequences [#\vec] of sharp functions on objects of type < \omega^CK₁ and of length < \omega^CK₁. These results use Borel Determinacy.

Let #^{1 \infty} = # \infty be the sharp function on all sets, and for k < \omega^CK₁, let #^{(k+1) \infty} (x) code indiscernibles for L(x)[[#\vec]^{k \infty}], where [#\vec]^{k \infty} = _df <#^{(\beta+1) \infty} | \beta < k >. In recent years, quasi-Borel Determinacy was used to show (*) when [#\vec] = # \infty and \Gamma(# \infty) = _df \Pi¹₁. In our talk, we present a generalization of this result; namely, (*) when [#\vec] = [#\vec]^{k \infty}, k < \omega^CK₁, and \Gamma([#\vec]^{k \infty}) = _df k * \Pi¹₁. We also define the relevant classes (e.g. ( \Gamma)^*₊, (k * \Gamma)).

Date received: March 18, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caac-11.