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Canonical measure assignments. (English) Zbl 1323.03071
Description analysis is a sophisticated machinery developed by the first author in his seminal work on the cardinal structures under determinacy assumptions [Lect. Notes Math. 1333, 117–220 (1988; Zbl 0677.03037); Mem. Am. Math. Soc. 670, 94 p. (1999; Zbl 0936.03045)]. In this paper, the authors present a relatively simple notational framework for describing the cardinal structure below the supremum of the projective ordinals, in order to avoid the description analysis.
The central notion of this framework is an ordinal algebra with an assignment of measures to its elements via two lifting operations (weak and strong lifting). The following known facts regarding the odd projective ordinals are used heavily throughout the paper:
A1: Each $$\pmb{\delta}^1_2$$ has the strong partition property: $$\pmb{\delta}^1_{2n+1} \to (\pmb{\delta}^1_{2n+1})^{\pmb{\delta}^1_{2n+1}}$$.
A2: Each $$\pmb{\delta}^1_2$$ is closed under ultrapowers.
A3: $$\pmb{\delta}^1_2 = \aleph_{\pmb{e}_n+1}$$.
The framework also needs a crucial technical assumption called canonicity assumption, which links the description analysis and the simplified presentation in this paper.
This framework is introduced in §3 and §4. In addition, in §5, the authors describe a recursive procedure for assigning measures. As an application of the canonicity assumption (see §6), the authors compute all regular cardinals below $$\aleph_{\varepsilon_0}$$, the cofinalities of the cardinals below $$\aleph_{\varepsilon_0}$$, and the Kleinberg sequences associated to all normal measures on the projective ordinals.
##### MSC:
 3e+55 Large cardinals 3e+60 Determinacy principles 300000 Other combinatorial set theory
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##### References:
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