SEALING OF THE UNIVERSALLY BAIRE SETS

G. Sargsyan, Nam Trang
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引用次数: 1

Abstract

Abstract A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. ${\sf Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The ${\sf Largest\ Suslin\ Axiom}$ ( ${\sf LSA}$ ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let ${\sf LSA}$ - ${\sf over}$ - ${\sf uB}$ be the statement that in all (set) generic extensions there is a model of $\sf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, $\sf {Sealing}$ is equiconsistent with $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that $\sf {Sealing}$ is weaker than the theory “ $\sf {ZFC}$ + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of $\sf {Sealing}$ by Woodin. A variation of $\sf {Sealing}$ , called $\sf {Tower \ Sealing}$ , is also shown to be equiconsistent with $\sf {Sealing}$ over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then $\sf {Sealing}$ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that $\sf {Sealing}$ holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ is not equivalent to $\sf {Sealing}$ .
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如果一组实数在拓扑空间中所有的连续原像都具有贝尔性质,则该实数集是普遍贝尔的。${\sf封合}$是Woodin引入的一种一般绝对条件,它以强有力的术语断言普遍贝尔集的理论不能被集合强迫改变。${\sf Largest\ Suslin\ Axiom}$ (${\sf LSA}$)是由Woodin分离出来的确定性公理。它断言对于有序可定义的抛射,最大的苏斯林基数是不可接近的。设${\sf LSA}$ - ${\sf /}$ - ${\sf uB}$为在所有(集)泛型扩展中存在一个$\sf {LSA}$的模型,其Suslin、cosuslin集是全称的Baire集。我们概述了在一些温和的大基数理论上,$\sf {sealed}$与$\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$是等价的。事实上,我们分离出一个精确的理论(在策略小鼠的层次结构中),它与两者是一致的(见定义3.1)。因此,我们得到$\sf {sealed}$弱于“$\sf {ZFC}$ +有一个Woodin基数,它是Woodin基数的一个极限”的理论。这大大改进了Woodin先前对$\sf{封口}$的一致性证明。$\sf{封口}$的一个变体,称为$\sf{塔\封口}$,也被证明与$\sf{封口}$在相同的大基本理论上是一致的。我们还概述了如果V有一个适当的Woodin基数类,一个强基数和一个一般普遍的Baire迭代策略,那么$\sf{封口}$在崩溃后最小强基数的后继数是可数的。这个结果是对前面提到的等一致性结果的补充,其中表明$\sf{封口}$在某个最小宇宙的一般扩展中成立。这个定理更普遍,因为它不需要最小假设。由此推论,$\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$并不等价于$\sf{封口}$。
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POUR-EL’S LANDSCAPE CATEGORICAL QUANTIFICATION POINCARÉ-WEYL’S PREDICATIVITY: GOING BEYOND A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF John MacFarlane, Philosophical Logic: A Contemporary Introduction, Routledge Contemporary Introductions to Philosophy, Routledge, New York, and London, 2021, xx + 238 pp.
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