Re: X
- From: Zaljohar@xxxxxxxxx
- Date: Fri, 7 Dec 2007 03:56:22 -0800 (PST)
On Dec 6, 5:05 pm, LauLuna <laureanol...@xxxxxxxx> wrote:
On Dec 6, 10:02 pm, Zaljo...@xxxxxxxxx wrote:
Hi
Let X be a theory in FOL with identity with the additional following
primitives: N,S,k,<.
N is a primitive one place predicate symbole that stands for 'natural
number'.
So N(x) stands for ' x is a natural number '.
S is primitive one place function symbole that stands for
'successor'.
So S(x) stands for the 'successor of x'.
k is a primitive constant.
< is a primitive two place relation symbole that stands for 'lesser
than'.
Axioms:
1) E!x( N(x) & ~Ey( x=S(y)) ).
Define x=0 <-> (N(x) & ~Ey( x=S(y))).
2) Ax ( N(x) -> N(S(x)) ).
3) Ax ( N(x) -> ~x=S(x) ).
4) Axyz( ( N(x) & N(y) & z=S(x) & z=S(y) ) -> x=y )
5) for any formula Q the closure of
(Q(0) &Ax((N(x) & Q(x)) -> Q(S(x)))) -> Ax ( N(x) ->Q(x) )
is an axiom.
6) Ax( N(x) -> x < S(x) ).
7) Axyz( (x<y & y<z) -> x<z )
8) ~ N(k)
9) k=S(k)
10) Ax ( N(x) -> x < k )
/
Theory definition finished.
Is this theory inconsistent?
Zuhair
No. Why should it? You have just added a sort of supernatural number
to the structure of the naturals.
What if Axiom 6 is modified to the following:
6) Ax( (N(x) v x=k) -> x < S(x) ).
Now from axiom 9 k=S(k) , then
from 6 and 9 we'll have k<k.
Now would this theory with this modification be inconsistent
or just counter-intutive?
Zuhair
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