PP-2006-09: Descriptions and cardinals below $\delta^1_5$

PP-2006-09: Jackson, Steve and Khafizov, Farid (2006) Descriptions and cardinals below $\delta^1_5$. [Report]

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We work throughout in the theory ZF+AD+DC. In the mid 80's, Jackson computed the values of the projective ordinals \delta^1_n. The upper bound in the general case appears in [J2], and the complete argument for \delta^1_5 appears in [J1]. We refer the reader to [Mo] or [Ke] for the definitions and basic properties of the \delta^1_n. A key part of the projective ordinal analysis is the concept of a description. Intuitively, a description is a finitary object "describing" how to build an equivalence class of a function f: \delta^1_3 \to \delta^1_3 with respect to certain canonical measures W^m_3 which we define below. The proof of the upper bound for the \delta^1_{2n+3} proceeds by showing that every successor cardinal less than \delta^1_{2n+3} is represented by a description, and then counting the number of descriptions. The lower bound for \delta^1_{2n+3} was obtained by embedding enough ultrapowers of \delta^1_{2n+1} (by various measures on \delta^1_{2n+1}) into \delta^1_{2n+3}. A theorem of Martin gives that these ultrapowers are all cardinals, and the lower bound follows. A question left open, however, was whether every description actually represents a cardinal. The main result of this paper is to show, below \delta^1_5 , that this is the case. Thus, the descriptions below \delta^1_5 exactly correspond to the cardinals below \delta^1_5 . Aside from rounding out the theory of descriptions, the results presented here also serve to simplify some of the ordinal computations of [J1]. In fact, implicit in our results is a simple (in principle) algorithm for determining the cardinal represented by a given description. This, in itself, could prove useful in addressing certain questions about the cardinals below the projective ordinals. The results of this paper are self-contained, modulo basic AD facts about \delta^1_1, \delta^1_3 which can be found, for example, in [Ke]. In particular, \delta^1_1=\omega_1, \delta^1_3=\omega_{\omega+1}, \delta^1_1 has the strong partition relation, and \delta^1_3 has the weak partition relation (actually, the strong relation as well, but we do not need this here). \omega, \omega_1, \omega_2 are the regular cardinals below \delta^1_3, and they, together with the c.u.b. filter, induce the three normal measures on \delta^1_3. Since we are not assuming familiarity with [J1], we present in the next section the definition of description and some related concepts. A few of our definitions are changed slightly from [J1]. We carry along through the paper some specific examples to help the reader through the somewhat technical definitions. In section 4 we give an application, and present a computational example.

Item Type: Report
Report Nr: PP-2006-09
Series Name: Prepublication (PP) Series
Year: 2006
Uncontrolled Keywords: projective ordinals; determinacy; ultrapowers
Depositing User: Benedikt
Date Deposited: 12 Oct 2016 14:36
Last Modified: 12 Oct 2016 14:36
URI: https://eprints.illc.uva.nl/id/eprint/184

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