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Higher Souslin trees and the GCH, revisited. (English) Zbl 1423.03169

Summary: It is proved that for every uncountable cardinal \(\lambda\), \(\mathrm{GCH} + \square(\lambda^+)\) entails the existence of a \(\mathrm{cf}(\lambda)\)-complete \(\lambda^+\)-Souslin tree. In particular, if GCH holds and there are no \(\aleph_2\)-Souslin trees, then \(\aleph_2\) is weakly compact in Gödel’s constructible universe, improving J. Gregory’s 1976 lower bound [J. Symb. Log. 41, 663–671 (1976; Zbl 0347.02044)]. Furthermore, it follows that if GCH holds and there are no \(\aleph_2\) and \(\aleph_3\) Souslin trees, then the Axiom of Determinacy holds in \(L(\mathbb{R})\).

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
03E55 Large cardinals

Citations:

Zbl 0347.02044
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Full Text: DOI

References:

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