函数投影与复杂度归约
Functional Projection and Complexity Reduction
AI总结：
针对现有复杂度归约证明依赖特定均匀化、难以提升至高阶射影层级的问题，引入弱于均匀化的函数投影原则，证明其可满足多数复杂度归约证明，并在可测基数的典范内模型中成立，进而得到相关组合族的复杂度归约结果。
摘要：
Törnquist（arXiv:1310.3250）证明，若存在$\boldsymbol{\Sigma}^1_2$极大几乎不交（MAD）族，则存在$\boldsymbol{\Pi}^1_1$ MAD族。此后，针对其他组合对象，如极大独立族或超滤子，也发现了若干其他“复杂度归约”证明。所有这些证明都依赖于$\boldsymbol{\Pi}^1_1$-均匀化，因此难以提升至高阶射影层级。本文引入一个弱于均匀化的原则，称之为“函数投影”，并证明该原则对大多数复杂度归约证明是充分的。另一方面，我们证明在$L[U]$（可测基数的典范内模型）中，函数投影对$\boldsymbol{\Pi}^1_2$关系成立。作为直接推论，我们得到在$L[U]$中，许多组合族存在$\boldsymbol{\Pi}^1_2$见证。
英文摘要：
Törnquist (arXiv:1310.3250) proved that if there exists a $\boldsymbolΣ^1_2$ maximal almost disjoint (MAD) family, then there exists a $\boldsymbolΠ^1_1$ MAD family. Since then, several other "complexity reduction" proofs have been found for other combinatorial objects, such as maximal independent families or ultrafilters. All of these proofs have relied on $\boldsymbolΠ^1_1$-uniformization, so they did not easily lift to higher projective levels. In this paper, we introduce a principle weaker than Uniformization which we call "Functional Projection", and show that this principle is sufficient for most complexity reduction proofs. On the other hand, we prove that in $L[U]$, the canonical inner model for a measurable cardinal, Functional Projection holds for $\boldsymbolΠ^1_2$ relations. As an immediate corollary, we obtain that in $L[U]$ there are $\boldsymbolΠ^1_2$ witnesses for many combinatorial families.
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