Quentin Anciaux skrev: > If you are ultrafinitist then by definition the set N does not > exist... (nor any infinite set countably or not). >
All sets are finite. It it (logically) impossible to construct an infinite set. You can construct the set N of all natural numbers. But that set must be finite. What the set N contains depends on how you have defined "natural number". > If you pose the assumption of a biggest number for N, you come to a > contradiction because you use the successor operation which cannot > admit a biggest number.(because N is well ordered any successor is > strictly bigger and the successor operation is always valid *by > definition of the operation*) > You have to define the successor operation. And to do that you have to define the definition set for that operation. So first you have to define the set N of natural numbers. And from that you can define the successor operator. The value set of the successor operator will be a new set, that contains one more element than the set N of natural numbers. This new element is BIGGEST+1, that is strictly bigger than all natural numbers. -- Torgny Tholerus > So either the set N does not exists in which case it makes no sense to > talk about the biggest number in N, or the set N does indeed exists > and it makes no sense to talk about the biggest number in N (while it > makes sense to talk about a number which is strictly bigger than any > natural number). > > To come back to the proof by contradiction you gave, the assumption > (2) which is that BIGGEST+1 is in N, is completely defined by the mere > existence of BIGGEST. If BIGGEST exists and well defined it entails > that BIGGEST+1 is not in N (but this invalidate the successor > operation and hence the mere existence of N). If BIGGEST in contrary > does not exist (as such, means it is not the biggest) then BIGGEST+1 > is in N by definition of N. > > Regards, > Quentin > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
