arXiv 2007.107642020-07-21math.LO

扩展逻辑的内部模型：第1部分

Inner Models from Extended Logics: Part 1

Juliette Kennedy, Menachem Magidor, Jouko Väänänen

摘要：

在哥德尔内部模型$L$的原始定义中，如果用二阶逻辑代替一阶逻辑，我们就得到HOD。在本文中，我们考虑的内部模型，如果我们取代一阶逻辑的逻辑，有一些，但不是全部，二阶逻辑的强度。典型的例子是一阶逻辑的广义量词的扩展，如Magidor-Malitz量词，共尾量词或稳态逻辑。我们的第一组结果表明，$L$和HOD都表现出一定量的{ \em形式主义自由}，在这个意义上，它们对底层逻辑的选择不是很敏感。我们的第二组结果表明，共尾量词产生了一个新的强大的内部模型之间的$L$和HOD。我们表明，除其他事项外，假设一个适当的类的Woodin基数的经常基数$>\aleph_1$的$V$是弱紧的内模型所产生的共尾量词和该模型的理论是（集）强制绝对和独立的共尾性的问题。我们不知道这个模型是否满足连续统假设，假设大基数，但我们可以证明，假设三个Woodin基数和一个在它们之上的可测，如果结构相对于一个实数，那么在一个实数锥上，连续统假设在相对化模型中是真的。

英文摘要：

If we replace first order logic by second order logic in the original definition of Gödel's inner model $L$, we obtain HOD. In this paper we consider inner models that arise if we replace first order logic by a logic that has some, but not all, of the strength of second order logic. Typical examples are the extensions of first order logic by generalized quantifiers, such as the Magidor-Malitz quantifier, the cofinality quantifier, or stationary logic. Our first set of results show that both $L$ and HOD manifest some amount of {\em formalism freeness} in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between $L$ and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals $>\aleph_1$ of $V$ are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals the Continuum Hypothesis is true in the relativized model.

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