THE GENERAL INDUCTIVE ARGUMENT FOR MEASURE ANALYSES WITH ADDITIVE ORDINAL ALGEBRAS

Transcript

1 THE GENERAL INDUCTIVE ARGUMENT FOR MEASURE ANALYSES WITH ADDITIVE ORDINAL ALGEBRAS STEFAN BOLD, BENEDIKT LÖWE In [BoLö ] we gave a survey of measure analyses under AD, discussed the general theory of order measures and gave a simple inductive argument for a measure analysis with just two measures that reached the first ω 2 cardinals after a strong partition cardinal In this article we show that the restriction to two measures can be lifted under the right prerequisites As a corollary we get a full measure analysis for sums of order measures This allows us to inductively reduce the measure analysis of an additive ordinal algebra A, with a set of generators V to the analysis of V The notion of an ordinal algebra is introduced in [JaLö ] This article is to be understood as a technical report, showing the progress we made in generalizing the results from [BoLö ], so we refer the reader to that paper for more detailed explanations and definitions of the notions and notations used in this paper 1 Necessary Lemmas and Definitions Lemma 1 Let κ < λ be cardinals, µ a measure on κ and cfλ > κ cfλ κ /µ = cfλ Proof Cf [BoLö, Lemma 6] Then Lemma 2 Let κ be a strong partition cardinal and let µ, η and ν be order measures on κ Then 1 κ κ /µ κ κ /µ ν, 2 κ κ /ν κ κ /µ ν, and 3 κ κ /µ ν κ κ /µ η ν Proof Cf [BoLö, Lemma 12] Let κ be a strong partition cardinal and let µ and ν be order measures, both on κ Let λ κ be a cardinal Then Proof Cf [BoLö, Lemma 13] λ κ /µ ν λ κ /ν κ /µ Theorem 4 Ultrapower Shifting Lemma Let β and γ be ordinals and let µ be a κ-complete ultrafilter on κ with κ κ /µ = κ γ If for all cardinals κ < ν κ β either ν is a successor and cfν > κ, or ν is a limit and cfν < κ, Date: January 11, Mathematics Subject Classification Primary 03E60, 03E05; Secondary 03E10 Both authors were funded by a DFG-NWO Bilateral Cooperation Grant DFG KO 1353/3-1; NWO DN

2 2 STEFAN BOLD, BENEDIKT LÖWE then κ β κ /µ κ γ+β Proof Cf [Lö02, Lemma 27] Definition 5 If µ i ; i < m + 1 is a finite sequence of measures on κ we write iult α µ 0,, µ m for the corresponding iterated Ultrapower: iult α µ 0,, µ m := α κ /µ m κ / κ /µ 0 We write iultµ 0,, µ m for iult κ µ 0,, µ m Definition 6 Let γ < ε 0 = sup n<ω e n be an ordinal and θ α ; α γ the unique sequence of ordinals such that θ 0 = 1, θ α+1 = θ α ω + 1 for α < γ and θ λ = sup α<λ θ α +1 for limit ordinals λ < γ For every successor ordinal ξ < sup α γ θ α ω the θ-cantor normal form of ξ is a decomposition of ξ into a finite sum of elements of θ α ; α γ, ie, ξ = θ α0 + + θ αm, where m ω It is defined by α 0 := min{α γ ; ξ < θ α+1 } and α i+1 := min{α γ ; ξ < θ α0 + + θ αi + θ α+1 } By wellfoundedness of < the relativized Cantor normal form of ξ is welldefined and unique The θ-cantor normal form of a limit ordinal ξ < sup α γ θ α ω is θ α0 + + θ αm 1, where θ α0 + + θ αm is the relativized Cantor normal form of ξ The abstract combinatorial computation Theorem 7 Let κ be a strong partition cardinal and γ < ε 0 an ordinal Let µ α ; α γ be a sequence of measures on κ and θ α ; α γ, ι α ; α γ sequences of ordinals such that i κ κ /µ 0 = κ θ0 = κ +, ii κ κ /µ α+1 = κ θα+1 = κ θα ω+1 for α < γ, iii κ κ /µ λ = κ θ λ = κ sup α<λ θα+1 for limit ordinals λ < γ, iv κ κ /ν θα κ κ /ν µ α for order measures ν and α < γ, and v cfκ κ /µ α = ι α > κ for α < γ Then for all ξ < sup α<γ θ α ω the following is true: 1 If ξ > 0 is a limit ordinal and θ α0 + + θ αm 1 its θ-cantor normal form then, with ζ = θ αm 1, κ ξ = iult κ ζµ α0,, µ αm 1 = κ ζ κ /µ α0 µ αm 1 2 If ξ is a successor ordinal and θ α0 + +θ αm its θ-cantor normal form then 3 κ ξ = iultµ α0,, µ αm = κ κ /µ α0 µ αm κ if ξ = 0, cfκ ξ := ω if ξ > 0 is a limit, ι αm if ξ = θ α0 + + θ αm is a successor Proof By assumption κ is a strong partition cardinal, thus regular Also, for all limit ordinals ξ < sup n<ω e n, the cofinality of κ ξ is ω So the first two parts of 3 are trivial

3 INDUCTIVE MEASURE ANALYSES 3 We proceed by induction on ξ > 0, using the following induction hypothesis: For all 0 < β ξ, the following three conditions hold: 1 If β is a limit and β + 1 = θ α0 + + θ αm, then κ β = iult κ ζµ α0,, µ αm 1, where ζ = θ αm 1 2 If β is a successor and β = θ α0 + + θ αm then κ β = iultµ α0,, µ αm = κ κ /µ α0 µ αm IH ξ 3 cfκ β := { ω if β > 0 is a limit, ι αm if β = θ α0 + + θ αm is a successor Obviously, if all IH ξ for ξ < sup α<γ θ α ω hold, the theorem is proven By assumption we have κ κ /µ 0 = κ + and cfκ + = cfκ κ /µ 0 = ι 0, so IH 1 holds For the successor step we assume that IH ξ holds and prove IH ξ+1 If ξ+1 = θ α for some α < γ, we have by assumption κ ξ+1 = κ κ /µ α and cfκ ξ+1 = cfκ κ /µ α = ι α and thus IH ξ+1 holds Otherwise let θ α0 + + θ αm be the θ-cantor normal form of ξ + 1 Then κ ξ+1 = κ θα 0 + +θα m 1 θαm = κ κ /µ α0 µ αm 1 θαm IH κ κ /µ α0 µ αm Assumption iv κ κ /µ αm κ /µ α0 µ αm 1 iultµ α0,, µ αm = iultµ α1,, µ αm κ /µ α0 = κ θα 1 + +θαm κ /µ α0 IH κ θα 0 + +θαm Theorem 4 = κ ξ+1 Using κ ξ+1 = iultµ α0,, µ αm 1 and Lemma 1 repeatedly we get ι αm 1 = cfκ κ /µ αm = = cfiultµ α1,, µ αm κ /µ α0 = cfκ ξ+1, which proves IH ξ+1 Now for the limit case We assume that IH β holds for all β < ξ and prove IH ξ Let θ α0 + + θ αm 1 be the θ-cantor normal form of ξ If α m is a successor we

4 4 STEFAN BOLD, BENEDIKT LÖWE have θ αm 1 = sup n ω θ αm 1 n and so we get κ ξ = sup n ω κ θα 0 + +θα m 1 +θαm 1 n = sup n ω κ κ /µ α0 µ αm 1 µ αm 1 n IH sup n ω κ κ /µ αm 1 n κ /µ α0 µ αm 1 = sup n ω κ θ αm 1 n [ κ /µ α0 µ αm 1 IH ] = κ θ αm 1 κ /µ α0 µ αm 1 sup n ω iultκ θ αm 1 nµ α0 µ αm 2, µ αm 1 sup n ω iultκ θ αm 1 nµ α0,, µ αm 1 [ = iultκ 1µ ] θαm α 0,, µ αm 1 sup n ω iult κ θα +θ m 1 αm 1 nµ α0,, µ αm 2 Theorem 4 sup n ω κ θ α0 + +θ αm 1 n Theorem 4 = κ ξ On the other hand, if α m 1 is a limit we have θ αm 1 = sup β αm θ β and so we get κ ξ = sup β αm κ θα 0 + +θα m 1 +θ β = sup β αm κ κ /µ α0 µ αm 1 µ β IH sup β αm κ κ /µ β κ /µ α0 µ αm 1 = sup β αm κ θ β [ κ /µ α0 µ αm 1 IH ] = κ θ αm 1 κ /µ α0 µ αm 1 sup β αm iultκ θ β µ α0 µ αm 2, µ αm 1 sup β αm iultκ θ β µ α0,, µ αm 1 [ = iultκ 1µ θαm α 0,, µ ] αm 1 sup β αm iult κ θα +θ m 1 β µ α0,, µ αm 2 Theorem 4 sup β αm κ θ α0 + +θ β Theorem 4 = κ ξ As we mentioned at the beginning of this proof, the cofinality of κ ξ for a limit ordinal 0 < ξ < ε 0 is ω, which concludes the proof Corollary 8 Let κ be a strong partition cardinal and γ < ε 0 an ordinal If µ α ; α γ is a sequence of measures on κ and θ α ; α γ, a sequence of ordinals that fulfill the requirements of Theorem 7, then for all ξ < sup α<γ θ α ω and finite sequences α i ; i m γ m+1 we have κ θα 0 + +θαm +ξ = iult κ ξµ α0,, µ αm = κ ξ κ /µα0 µ αm Proof If θ β0 + + θ βn is the θ-cantor normal form of θ α0 + + θ αm + ξ, then the θ-cantor normal form of ξ is an end segment of θ β0 + + θ βn, ie there is a k 0 such that θ βk + + θ βn is the θ-cantor normal form of ξ And for all i < k

5 INDUCTIVE MEASURE ANALYSES 5 there is a j m such that θ βi = θ αj So by Theorem 7 κ θα 0 + +θαm +ξ = κ θ β 0 + +θ βn = κ κ /µ β0 µ βn, using Lemma 2 we can insert the missing elements of the sequence µ αi ; i m and then apply to get κ κ /µ β0 µ βn κ κ /µ α0 µ αm µ βk µ βn κ κ /µ βk µ βn κ /µ α0 µ αm = κ ξ κ /µα0 µ αm And finally we can use and Theorem 4, both repeatedly as we did before in the proof of Theorem 7, to reach equality: κ ξ κ /µα0 µ αm iult κ ξµ α0,, µ αm κ θα 0 + +θαm +ξ References [BoLö ] [JaLö ] [Lö02] Stefan Bold and Benedikt Löwe, A simple inductive measure analysis for cardinals under the Axiom of Determinacy, submitted to the Proceedings of the North Texas Logic Conference; ILLC Publication Series PP Steve Jackson, Benedikt Löwe, Canonical Measure Assignments in preparation Benedikt Löwe, Kleinberg Sequences and partition cardinals below δ 1 5, Fundamenta Mathematicae , p S Bold & B Löwe Institute for Logic, Language and Computation, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands address: {sbold, bloewe}@scienceuvanl S Bold & B Löwe Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Beringstraße 1, Bonn, Germany address: {bold, loewe}@mathuni-bonnde