A Euclidean comparison of the sizes of sets
speaker: Marco Forti (Università di Pisa)abstract: The usual way of comparing and evaluating the sizes of sets is the “Cantorian” way, grounded on the so called Hume’s principle “two sets have equal size if and only if they are equipotent”, thus violating the fundamental Euclid’s principle “the whole is greater than the part”.
The Cantorian way has been deeply investigated since the very birth of set theory; here we consider instead a “Euclidean” way, introduced at the beginning of the century, that preserves the natural idea that a set is (strictly) larger than its proper subsets, still maintaining the weak
Hume’s principle “sets having equal sizes are equipotent”, and, more rel- evant, the Cantorian definitions of order, addition, and multiplication of
sizes, through subsets, (disjoint) unions, and cartesian products. We introduce a total preordering of the universe of sets whose associated equivalence
X ' Y ⇐⇒ X Y X satisfies the natural set theoretic condition X ≺ Y ⇐⇒ ∃Z X ' Z ⊂ Y.
(Remark the use of strict preordering corresponding to proper inclusion:
this “proper subset property”, initially proved consistent only for count- able sets, has been recently extended to sets of any cardinality, thus an- swering one of the main open question in this area.)
This Euclidean notion of size of sets (numerosity), given by the equiv- alence classes modulo ', satisfies all the five Euclid’s common notions on
magnitudines, and gives rise to an ordered semiring of nonstandard natu- ral numbers, thus enjoying the best arithmetic, to be contrasted with the
aukward arithmetic of infinite cardinals, where a+b = a · b = max {a, b}.
timetable:
Thu 12 Sep, 12:10 - 12:45, Aula Dini
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