论文标题
集合大小的欧几里得比较理论
A Euclidean comparison theory for the size of sets
论文作者
论文摘要
我们讨论了比较和评估集合大小的两种主要方法：以所谓的休ume原理为基础的“ cantorian”方式（如果有等值的话，两个组的大小相等），以及“欧几里得”的方式，保持欧几里得的原理“整体大于整个部分”。自集体理论的诞生以来，我们对前者进行了深入的研究，我们将其集中在大小（数字）的“欧几里得”概念上，该概念维持了秩序，加法和乘法的cantorain偏差，同时保留了（严格地）比其适当的子集更大的（严格）集合的自然观念。 这些数字满足了五个欧几里得的共同概念，并构成了非标准自然数的半含量，因此享受了最好的算术。最相关的是套装的自然集理论定义}： $$ x \ prec y \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \ simeq z \ simeq z \ subset y $$从本世纪初从可数到无法数的集合扩展了``适当的子集属性''从可数到无数的集合一直是该领域的主要开放问题。
We discuss two main ways in comparing and evaluating the size of sets: the "Cantorian" way, grounded on the so called Hume principle (two sets have equal size if they are equipotent), and the "Euclidean" way, maintaining Euclid's principle "the whole is greater than the part". The former being deeply investigated since the very birth of set theory, we concentrate here on the "Euclidean" notion of size (numerosity), that maintains the Cantorain defiitions of order, addition and multiplication, while preserving the natural idea that a set is (strictly) larger than its proper subsets. These numerosities satisfy the five Euclid's common notions, and constitute a semiring of nonstandarda natural numbers, thus enjoying the best arithmetic. Most relevant is the natural set theoretic definition} of the set-preordering: $$X\prec Y\ \ \Iff\ \ \exists Z\ X\simeq Z\subset Y$$ Extending this ``proper subset property" from countable to uncountable sets has been the main open question in this area from the beginning of the century.
