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| zuhair... |
 Posted: Fri Nov 06, 2009 2:09 pm |
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Guest
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See the following rather simple set theory in first order logic with
identity and epsilon, with the following Axioms:
1.Extensionality
2.Foundation
3.Empty
4.Comprehension: for n=0,1,2,3,....., If Phi(y,x1,
,xn) is a formula
in which at least y is free with parameters x1,...,xn, and in which c
is not free, and Phi(z,x1,
,xn) is exactly the formula obtained by
replacing all occurrences of y in Phi(y,x1,...,xn) by the symbol z,
then
~ [for all x1,
,xn
(for all y (Phi(y,x1,
,xn) -> Exist z (yez & Phi(z,x1,
,xn))))]
->
for all x1,
,xn Exist c For all y (yec <-> Phi(y,x1,
,xn))
are axioms.
5.Infinity
were 1.2.3.5. all as in ZF set theory.
Theory definition finished/
Now this theory must be inconsistent! but what is the proof of that
inconsistency?
Zuhair |
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| zuhair... |
| Posted: Sun Nov 08, 2009 4:26 am |
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Guest
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On Nov 6, 7:09 pm, zuhair <zaljo... at (no spam) gmail.com> wrote:
[quote]See the following rather simple set theory in first order logic with
identity and epsilon, with the following Axioms:
1.Extensionality
2.Foundation
3.Empty
4.Comprehension: for n=0,1,2,3,....., If Phi(y,x1,
,xn) is a formula
in which at least y is free with parameters x1,...,xn, and in which c
is not free, and Phi(z,x1,
,xn) is exactly the formula obtained by
replacing all occurrences of y in Phi(y,x1,...,xn) by the symbol z,
then
~ [for all x1,
,xn
(for all y (Phi(y,x1,
,xn) -> Exist z (yez & Phi(z,x1,
,xn))))]
-
for all x1,
,xn Exist c For all y (yec <-> Phi(y,x1,
,xn))
are axioms.
5.Infinity
were 1.2.3.5. all as in ZF set theory.
Theory definition finished/
Now this theory must be inconsistent! but what is the proof of that
inconsistency?
Zuhair
[/quote]
This theory is inconsistent, its actually very easy to prove that!
Zuhair |
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