Bruno Marchal skrev: > 0) Bijections > > Definition: A and B have same cardinality (size, number of elements) > when there is a bijection from A to B. > > Now, at first sight, we could think that all *infinite* sets have the > same cardinality, indeed the "cardinality" of the infinite set N. By N, > I mean of course the set {0, 1, 2, 3, 4, ...} > What do you mean by "..."? > By E, I mean the set of even number {0, 2, 4, 6, 8, ...} > > Galileo is the first, to my knowledge to realize that N and E have the > "same number of elements", in Cantor's sense. By this I mean that > Galileo realized that there is a bijection between N and E. For > example, the function which sends x on 2*x, for each x in N is such a > bijection. > What do you mean by "each x" here?

How do you prove that each x in N has a corresponding number 2*x in E? If m is the biggest number in N, then there will be no corresponding number 2*m in E, because 2*m is not a number. > Now, instead of taking this at face value like Cantor, Galileo will > instead take this as a warning against the use of the infinite in math > or calculus. > -- Torgny Tholerus --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---