2022/12/11 by Marco Forti, Forti, Marco
Computer Science · Mathematics · #03A05 (secondary) #03C20 #03E65 (primary) #Computability, Logic, AI Algorithms #FOS: Mathematics #History and Theory of Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2212.05527
openalex publication_date 2022/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∃ Z X≃ Z⊂ 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.