apoorv
Posts:
88
Registered:
6/18/05


Re: Is the set N of natural numbers well defined?
Posted:
Dec 29, 2005 6:36 AM


The question whether N belongs to N is really the same as whether the axiom of infinity is consistent with the axiom of foundation.
As per the axiom of infinity, N={0}U {Sx: x e N}
Now, the axiom of foundation requires that each x satisfy the condition Sx !=x, or equivalently, the condition x !e x. (In fact, x!ex <>Sx!=x). Then, N={0}U{Sx: x e N, and x!ex} Also, Sx:x e N, <>y:y e N, y!=0 And, x:x!ex <>Sx:Sx !e Sx <>y:y !e y. Then,
N={0}U{y:yeN, y!=0,and y !e y }, Or,N is the set containing 0 and those sets which are in N and do not contain themselves. The condition y e N is tautological and the only constraint is y!e y. So,
N={0} U {y:y !=0 ,and y !e y}.
In other words, the axiom of infinity implies that N is precisely the set of those ordinals that do not contain themselves. We know that this set is not well defined.
Apoorv

