论文标题
小的可衡量的红衣主教
Small measurable cardinals
论文作者
论文摘要
我们继续从[5]开始工作，并对$ j $ - 多功能系统的定义进行了较小但显着的改进。这为我们提供了更好地提起基本嵌入到对称扩展中。特别是，这使我们更容易提起弱紧密的嵌入，从而保留弱关键的红衣主教的概念。我们使用这种改进的举重标准表明，第一个可测量的基数可以是第一个弱关键的红衣主教或第一个Mahlo Cardinal，这都是相对于单个可测量的基数的存在。但是，如果第一个无法访问的红衣主教是第一个可测量的红衣主教，则在合适的内部型号中，它的米切尔订单至少为$ 2 $。
We continue to work from [5] and make a small -- but significant -- improvement to the definition of $j$-decomposable system. This provides us with a better lifting of elementary embeddings to symmetric extension. In particular, this allows us to more easily lift weakly compact embedding and thus preserve the notion of weakly critical cardinals. We use this improved lifting criterion to show that the first measurable cardinal can be the first weakly critical cardinal or the first Mahlo cardinal, both relative to the existence of a single measurable cardinal. However, if the first inaccessible cardinal is the first measurable cardinal, then in a suitable inner model it has a Mitchell order of at least $2$.
