Monotone versus non-monotone projective operators

IF 0.9 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-12-17 DOI:10.1112/blms.13194

J. P. Aguilera, P. D. Welch

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Abstract

For a class of operators $Γ$ , let $| Γ |$ denote the closure ordinal of $Γ$ -inductive definitions. We give upper bounds on the values of $| Σ_{2 n + 1}^{1, m o n} |$ and $| Π_{2 n + 2}^{1, m o n} |$ under the assumption that all projective sets of reals are determined, significantly improving the known results. In particular, the bounds show that $| Π_{n}^{1, m o n} | < | Π_{n}^{1} |$ and $| Σ_{n}^{1, m o n} | < | Σ_{n}^{1} |$ hold for $2 ⩽ n$ under the assumption of projective determinacy. Some of these inequalities were obtained by Aanderaa in the 70s via recursion-theoretic methods but never appeared in print. Our proofs are model-theoretic.

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单调与非单调射影算子

对于一类操作符Γ $\Gamma$，让| Γ | $|\Gamma |$表示Γ $\Gamma$归纳定义的闭包序数。我们给出| Σ 2 n + 1 1值的上界，M o n | $|\Sigma ^{1,mon}_{2n+1}|$和| Π 2 n+ 21, m on | $|\Pi ^{1,mon}_{2n+2}|$假设所有实数的投影集都已确定，显著改善已知结果。特别是，边界显示| Π n 1，M o n | ＆lt；| Π n 1 | $|\Pi ^{1,mon}_{n}| < |\Pi ^1_{n}|$和| Σ n1、我不知道你在说什么；| Σ n1| $|\Sigma ^{1,mon}_{n}| < |\Sigma ^1_{n}|$在射影确定性假设下保持2≤n $2\leqslant n$。其中一些不等式是Aanderaa在70年代通过递归理论方法得到的，但从未在印刷品中出现过。我们的证明是模型论的。

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