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Topic: Is the set N of natural numbers well defined?
Replies: 67   Last Post: Feb 10, 2006 2:21 AM

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 apoorv Posts: 89 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

Date Subject Author
6/18/05 apoorv
6/19/05 fernando revilla
6/22/05 apoorv
6/22/05 Martin Semerád
6/28/05 apoorv
6/28/05 Virgil
6/29/05 apoorv
6/28/05 fernando revilla
6/29/05 Martin Semerád
6/29/05 fernando revilla
6/29/05 Martin Semerád
6/29/05 fernando revilla
6/29/05 Martin Semerád
6/30/05 apoorv
6/30/05 Dik T. Winter
7/1/05 Martin Semerád
7/1/05 Dik T. Winter
7/1/05 apoorv
7/4/05 apoorv
7/4/05 Timothy Little
7/8/05 apoorv
7/8/05 Timothy Little
7/9/05 apoorv
7/9/05 Timothy Little
7/11/05 apoorv
7/11/05 Timothy Little
7/12/05 apoorv
8/16/05 apoorv
8/17/05 fernando revilla
8/23/05 apoorv
9/1/05 apoorv
9/2/05 Timothy Little
9/3/05 apoorv
9/3/05 Timothy Little
9/4/05 apoorv
9/4/05 Timothy Little
9/6/05 apoorv
9/6/05 Timothy Little
9/11/05 apoorv
9/11/05 David Kastrup
9/11/05 Timothy Little
9/14/05 apoorv
9/15/05 Timothy Little
9/21/05 apoorv
9/21/05 Timothy Little
9/23/05 apoorv
9/23/05 Timothy Little
9/24/05 apoorv
9/24/05 Timothy Little
9/26/05 apoorv
9/26/05 Rainer Rosenthal
10/2/05 apoorv
10/2/05 Virgil
10/3/05 apoorv
9/21/05 apoorv
9/21/05 Timothy Little
9/24/05 apoorv
9/24/05 Timothy Little
9/26/05 apoorv
10/12/05 apoorv
11/7/05 fernando revilla
1/9/06 apoorv
12/29/05 apoorv
1/2/06 apoorv
1/2/06 Virgil
1/3/06 apoorv
2/10/06 apoorv
1/9/06 apoorv