PP200609: Jackson, Steve and Khafizov, Farid (2006) Descriptions and cardinals below $\delta^1_5$. [Report]

Text (Full Text)
PP200609.text.pdf Download (218kB)  Preview 

Text (Abstract)
PP200609.abstract.txt Download (2kB) 
Abstract
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 selfcontained, 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:  PP200609 
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 
Actions (login required)
View Item 