Le 14-nov.-07, à 17:23, Torgny Tholerus a écrit :

> > 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 "..."? Are you asking this as a student who does not understand the math, or as a philospher who, like an ultrafinist, does not believe in the potential infinite (accepted by mechanist, finistist, intuitionist, etc.). I have already explained that the meaning of "...'" in {I, II, III, IIII, IIIII, IIIIII, IIIIIII, IIIIIIII, IIIIIIIII, ...} is *the* mystery. A beautiful thing, which is premature at this stage of the thread, is that accepting the usual meaning of "..." , then we can mathematically explained why the meaning of "..." has to be a mystery. >> 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? I mean "for each natural number". > > 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, There is no biggest number in N. By definition of N we accept that if x is in N, then x+1 is also in N, and is different from x. > then there will be no corresponding > number 2*m in E, because 2*m is not a number. Of course, but you are not using the usual notion of numbers. If you believe that the usual notion of numbers is wrong, I am sorry I cannot help you. Bruno >> 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 > > > > http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---