强迫公理和非定常理想的复杂性。

Pub Date : 2022-01-01 Epub Date: 2022-06-27 DOI:10.1007/s00605-022-01734-w

Sean Cox, Philipp Lücke

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引用次数: 0

摘要

研究了强强迫公理对ω 2上非平稳理想的复杂性的影响及其对某些伴随性的限制。我们的主要结果表明，马丁极大值的增强并不能决定ω 2上的非平稳理想对可数共度序数集的约束是否为Δ 1 -可由H (ω 3)中的参数公式定义。在证明这一结果中发展的技术也使我们能够证明与CH兼容的ω 2上的完全非平稳理想和强强迫公理的类似结果。最后，我们回答了S. Friedman, Wu和zdomsky的一个问题，证明了ω 2上的非平稳理想的Δ 1 -可定义性与ω 2上任意大的连续统函数相容。

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Forcing axioms and the complexity of non-stationary ideals.

We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $ω_{2}$ and its restrictions to certain cofinalities. Our main result shows that the strengthening ${MM}^{+ +}$ of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on $ω_{2}$ to sets of ordinals of countable cofinality is $Δ_{1}$ -definable by formulas with parameters in $H (ω_{3})$ . The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $ω_{2}$ and strong forcing axioms that are compatible with $CH$ . Finally, we answer a question of S. Friedman, Wu and Zdomskyy by showing that the $Δ_{1}$ -definability of the non-stationary ideal on $ω_{2}$ is compatible with arbitrary large values of the continuum function at $ω_{2}$ .

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