Le 20-nov.-07, à 12:14, Torgny Tholerus a écrit :
> > Bruno Marchal skrev: >> >> To sum up; finite ordinal and finite cardinal coincide. Concerning >> infinite "number" there are much ordinals than cardinals. In between >> two different infinite cardinal, there will be an infinity of ordinal. >> We have already seen that omega, omega+1, ... omega+omega, >> omega+omega+1, ....3.omega, ... 4.omega .... ....omega.omega ..... >> omega.omega.omega, .....omega^omega ..... are all different ordinals, >> but all have the same cardinality. >> > Was it not an error there? 2^omega is just the number of all subsets > of > omega, and the number of all subsets always have bigger cardinality > than > the set. Yes, that is true. > So omega^omega can not have the same cardinality as omega. But addition, multiplication, and thus exponentiation are not the same operation for ordinals and cardinals. I should have written omega"^"omega, or something like that. That is why I have written 3.omega instead of 3*omega. We can come back on ordinal later, but now I will focus the attention on the cardinals, and prove indeed that 2^omega, or 2^N, or equivalently the infinite cartesian product (of sets) 2X2X2X2X2X2X2X2X... , is NOT enumerable (and indeed vastly bigger that the ordinal omega"^"omega. You can look at the thread on the growing functions for a little more on the ordinals. Actually my point was to remind people of the difference between ordinal and cardinal, and, yes, they have different addition, multiplication, etc. Bruno 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 -~----------~----~----~----~------~----~------~--~---
