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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:
03E55 Large cardinals
03E60 Determinacy principles
03E05 Other combinatorial set theory
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References:
[1] Memoirs of the American Mathematical Society 140 (1999)
[2] DOI: 10.1007/BFb0084974
[3] Universal algebra (1968)
[4] DOI: 10.1090/conm/425/08115
[5] Handbook of set theory 3 pp 1753– (2010)
[6] Infinitary combinatorics and the axiom of determinateness 612 (1977) · Zbl 0362.02067
[7] DOI: 10.4064/fm171-1-4 · Zbl 0999.03047
[8] DOI: 10.1007/BFb0069296
[9] The higher infinite, large cardinals in set theory from their beginnings (1994) · Zbl 0813.03034
[10] Descriptive set theory 100 (1980) · Zbl 0433.03025
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