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Register a new userDetermined Admissible Sets
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Authors
Juan P. Aguilera
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It is shown, from hypotheses in the region of $omega^2$ Woodin cardinals, that there is a transitive model of KP + AD$_mathbb{R}$ containing all reals.
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Harrington and Soare introduced the notion of an n-tardy set. They showed that there is a nonempty $mathcal{E}$ property Q(A) such that if Q(A) then A is 2-tardy. Since they also showed no 2-tardy set is complete, Harrington and Soare showed that there exists an orbit of computably enumerable sets such that every set in that orbit is incomplete. Our study of n-tardy sets takes off from where Harrington and Soare left off. We answer all the open questions asked by Harrington and Soare about n-tardy sets. We show there is a 3-tardy set A that is not computed by any 2-tardy set B. We also show that there are nonempty $mathcal{E}$ properties $Q_n(A)$ such that if $Q_n(A)$ then A is properly n-tardy.
We prove that a wide Morley sequence in a wide generically stable type is isometric to the standard basis of an $ell_p$ space for some $p$.
Soare proved that the maximal sets form an orbit in $mathcal{E}$. We consider here $mathcal{D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer. Some orbits of $mathcal{D}$-maximal sets are well understood, e.g., hemimaximal sets, but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the $mathcal{D}$-maximal sets. Although these invariants help us to better understand the $mathcal{D}$-maximal sets, we use them to show that several classes of $mathcal{D}$-maximal sets break into infinitely many orbits.
We prove that for every Scott set $S$ there are $S$-saturated real closed fields and models of Presburger arithmetic.
We prove a number of results on the determinacy of $sigma$-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between $sigma$-projective determinacy and the determinacy of certain classes of games of variable length ${<}omega^2$ (Theorem 2.4). We then give an elementary proof of the determinacy of $sigma$-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof to obtain proofs of the determinacy of $sigma$-projective games of a given countable length and of games with payoff in the smallest $sigma$-algebra containing the projective sets, from corresponding assumptions (Theorems 5.1 and 5.4).
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