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
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
> 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.
> 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.
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