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Canonical measure assignments

Steve Jackson and Benedikt Löwe

The Journal of Symbolic Logic / Volume 78 / Issue 02 / June 2013, pp 403 - 424

DOI: 10.2178/jsl.7802040, Published online: 12 March 2014

Link to this article: http://journals.cambridge.org/abstract_S0022481200000062

How to cite this article:

Steve Jackson and Benedikt Löwe (2013). Canonical measure assignments . The Journal of

Symbolic Logic, 78, pp 403-424 doi:10.2178/jsl.7802040

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 78, Number 2, June 2013

CANONICAL MEASURE ASSIGNMENTS

STEVE JACKSON AND BENEDIKT LOWE

Abstract. We work under the assumption of the Axiom of Determinacy and associate a measure to each cardinal K < N£j) in a recursive definition of a canonical measure assignment. We give algorithmic applications of the existence of such a canonical measure assignment (computation of cofinalities, computation of the Kleinberg sequences associated to the normal ultrafilters on all projective ordinals). §1. Introduction. One of the striking features of set theory under the Axiom of

Determinacy is the fact that there is a full analysis of the cardinal structure for a fairly large initial segment of© := sup{a : there is a surjection from M onto a}, something which we cannot hope to get in the ZFC context. While almost none of the combinatorial properties of small cardinals (e.g., N2, K3, Hm2) are fixed in ZFC,

ZF+AD gives us definite combinatorial properties (e.g., measurability, Jonssonness,

Rowbottomness) of these cardinals, in particular below H£o, the supremum of the projective ordinals (defined in §2).

This structure is closely tied to an analysis of measures on the projective ordinals and the representation of cardinals as ultrapowers via these measures. In [4] this analysis is given below Sl5, and in [3] it is extended to all the Sn. A key combinatorial ingredient in this analysis is the notion of a description, which gives a precise presentation of the cardinal structure (see also [5] for an introduction to this theory).

In the paper [6] which can be seen as a companion to this one, it was shown that a certain fairly simple family of measures on 8 \ could be used to directly describe the cardinal structure below d\. This presentation of the cardinal structure avoided the notion of description, although the description theory was an integral part of the proofs.

Our goal here is to present a simple combinatorial framework which suffices to describe the cardinal structure below the supremum of the projective ordinals, and which also avoids the description analysis. We introduce the notion of an ordinal algebra, and we inductively assign measures to the elements of this algebra through two lifting operations. This gives us a comparatively simple notational framework

Received March 23, 2010. 2010 Mathematics Subject Classification. 03E60, 03E55 (primary); 03E05 (secondary).

The first author was supported by NSF Grant DMS-0097181; the second author's visits to Denton

TX were made possible by two DFG-NWO Bilateral Cooperation Grants (DFG KO 1353/3-1 & KO 1353/5-1, NWO 61-532 & 62-630), an NWO reisbeurs (R-62-616) and the Roy McLeod Millican

Memorial Fund. Work of the authors done at the Erwin-Schrodinger-Institut in Vienna in June 2009 was partially supported by ESF grant PESC/SCHINFTY2898 of the Network INFTY. © 2013, Association for Symbolic Logic

OO22-4812/13/7802-0OO4/S3.20

DOI:10.2178/jsl.7802040 403 404 STEVE JACKSON AND BENEDIKT LOWE for describing the cardinal structure below the 6„ which is of independent interest and will also allow those not familiar with the description analysis to use many of the strong consequences of that theory.

Throughout this paper we make the following standing assumptions about the odd projective ordinals (definitions are given in § 2):

Al: Each^jn+i has the strong partition property: d\n+x -> (^n+i)'*2"*1A2: Each S\n+l is closed under ultrapowers.

A 3 : <*L+i = Nc„+i.t

We emphasize that we do not prove the assumptions Al, A2, or A3 here, although we use these properties heavily. Assumption Al is one of the central results of the description analysis. A proof using descriptions for 8 x = co\ can be found in [5] (the original proof is due to Martin) and for S\ = a>w+\ can be found in [4]. Also, [3] extends the description theory to all levels of the projective hierarchy (though the strong partition property for the general Sl2n+l is not explicitly proved in [3], it is implicit from the analysis of [3] and the arguments of [4]). Assumption A2 follows from the existence of a &\n+\ coding of the subsets of k\n+{, the cardinal predecessor of d\n+l. For n = 0 this is trivial, for n — 1 a proof is given in [5] (the original proof is due to Kunen), for n = 2 a proof is given in [4], and [3] again gives the machinery for generalizing the proof of [4] to arbitrary levels. Assumption A3 is the main result of [3] (cf. below).

Aside from these general standing assumptions on the S\n+1, there is a crucial technical assumption we shall eventually make which we call the canonicity assumption (defined precisely later). This assumption concerns the values of certain ultrapowers of the S\n+l by certain measures on these ordinals which we introduce.

The canonicity assumption is perhaps not as intuitive as the above standing assumptions, but it is an important technical link between the description theory and the more simplified representation of the current paper. A proof of the canonicity assumption for the measures on 3 \ using the description analysis is given in [6]. With the general arguments of [3], this canonicity proof should extend to higher levels, but this will appear later. Our purpose here is show how these assumptions lead to a simpler and elegant way to describe the cardinal structure below the supremum of the projective ordinals (which is the cardinal KEo; see Theorem 4).