[FOM] None

stevnewb@ix.netcom.com stevnewb at ix.netcom.com
Fri Feb 7 18:10:30 EST 2003

A question which has nothing to do with predicativity:

Suppose we have a cwff ~B  which has an infinite, but no finite models. [It 
can be the
conjunction of the axioms for any theory that has no finite realizations.]

Then  B  is n-valid [=df= valid on all and only finite domains.] 
If  B  is  **valid** on all
finite domains then B  has models on all finite domains. The union of those 
models would have to be an infinite model. [I'm not invoking compactness 
here, merely the existence of unions.]

That would entail that both B , ~B  have infinite models. In all my reading 
in logic over the past forty years, I can't recall ever having come across 
anything that either asserted or denied the possibility of this "paradox". 
Or even mentioned it.

Suppose that the theory in question is the perennial favorite, Peano's 
Arithmetic. The axioms in the conjunction would be arranged so that 
the  **last** axiom is

"A:(n)[ n' /= 0] "

[Read 'A:' and  'E:' as universal and existential quantifiers.]

Then  B  replaces the last conjunction ' & ' with '=>' and negates the last 
axiom, so we have

                         [&(first n-1 axioms)] => E:(n)[ n' = 0 ].

In the finite models there is no problem, but how to interpret this 
distinguished n'? Perhaps as the smallest inaccessible cardinal?

This is not a put-on, I'm asking a serious question to which I do not know 
the answer.



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