Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap. Abstractionists about numbers—those who think natural numbers are objects characterized by abstraction principles—face an acute version of this problem. Many abstraction principles appear to characterize the natural numbers. If each abstraction principle determines its own sort, then there is no single subject-matter of arithmetic—there are too many numbers. That is, unless objects can belong to more than one sort. But if there are multi-sorted objects, there should be cross-sortal identity principles for identifying objects across sorts. The going cross-sortal identity principle, ECIA2 of (Cook and Ebert 2005), solves the problem of too many numbers. But, I argue, it does so at a high cost. I therefore propose a novel cross-sortal identity principle, based on embeddings of the induced models of abstracts developed by Walsh (2012). The new criterion matches ECIA2’s success, but offers interestingly different answers to the more controversial identifications made by ECIA2.

中文翻译：

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识别有限的基数摘要

对象似乎分为不同种类，每种都有自己的身份标准。这就提出了排序是否重叠的问题。关于数字的抽象主义者——那些认为自然数是以抽象原则为特征的对象的人——面临着这个问题的尖锐版本。许多抽象原则似乎可以表征自然数。如果每个抽象原则都决定了它自己的类型，那么算术就没有单一的主题——数字太多了。也就是说，除非对象可以属于多个种类。但是如果有多重排序的对象，应该有跨排序识别对象的跨排序标识原则。(Cook and Ebert 2005) 的 ECIA2 跨排序恒等式原则解决了数字过多的问题。但是，我认为，这样做的成本很高。因此，我基于 Walsh (2012) 开发的摘要诱导模型的嵌入，提出了一种新的交叉分类身份原则。新标准与 ECIA2 的成功相匹配，但对 ECIA2 做出的更具争议性的识别提供了有趣的不同答案。