论文标题
强迫无选的模型和通用绝对性
Forcing over choiceless models and generic absoluteness
论文作者
论文摘要
我们开发了一个工具箱，以强迫无需公理的设定理论的任意模型。特别是，我们引入了可数链条件的变体，并证明了迭代定理适用于许多经典强迫，例如Cohen强迫和随机代数。我们的方法避开了一个问题，即通过Karagila和Schweber的最新结果，使用可数链条条件可能会崩溃$ω_1$。使用此过程，我们表明，添加许多Cohen Real和随机实物会导致不同的理论。这个结果是由于伍丁。因此，人们总是可以通过强迫来改变宇宙理论，就像连续假设一样，可以通过强迫选择选择的任意模型来获得否定。我们进一步研究了规定的原则，表明宇宙的一阶理论在所有通用延伸中都通过固定类别的强迫保持不变。扩展了伍丁的结果，我们表明，即使对于非常有限的类别，例如Cohen强迫的所有有限支持产品或所有随机代数的类别的类别，该原理也意味着所有无限的红衣主教都具有可计数的辅助性。
We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration theorem that applies to many classical forcings such as Cohen forcing and random algebras. Our approach sidesteps the problem that forcing with the countable chain condition can collapse $ω_1$ by a recent result of Karagila and Schweber. Using this, we show that adding many Cohen reals and random reals leads to different theories. This result is due to Woodin. Thus one can always change the theory of the universe by forcing, just like the continuum hypothesis and its negation can be obtained by forcing over arbitrary models with choice. We further study principles stipulating that the first-order theory of the universe remains the same in all generic extension by a fixed class of forcings. Extending a result of Woodin, we show that even for very restricted classes such as the class of all finite support products of Cohen forcing or the class of all random algebras, this principle implies that all infinite cardinals have countable cofinality.
