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

Preview |
Text (Full Text)
PP-2006-09.text.pdf Download (218kB) | Preview |

Text (Abstract)
PP-2006-09.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 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 |

## Actions (login required)

View Item |