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george says...
>I am sad about the fact that you seem not to know
>a universally quantified variable when you see one,
>and I am even sadder that ranting is the only thing I
>can do about it.
George, you are completely mistaken here. What is
going on is that Peter is *describing* a language
whose axioms have no variables (free or otherwise).
He is *using* a different language to describe this
first language. This second language is the *metalanguage*.
Metalanguage is used to talk about an object language.
There are no variables in the object language, but there
*are* variables in the metalanguage. I know that's a
little hard to understand, but that isn't Peter's fault.
Once again, Peter is *describing* a language with no
variables occurring in any of the axioms. The axioms
of this language includes formulas such as
not (0 = S(0))
not (0 = S(S(0)))
not (0 = S(S(S(0))))
not (0 = S(S(S(S(0)))))
Since Peter doesn't have the time to write down the
entire infinite list of axioms, he is instead using
a shorthand, under the (false, in your case) assumption
that people reading this newsgroup will understand that
shorthand.
He summarizes the entire infinite list of axioms by
giving the pattern
not (0 = S(m))
Any instance of the infinite list of axioms can
be obtained from this pattern by substituting
a closed term for m. The pattern is *not* an
axiom of the object language. It's Peter's way
of telling you, in a finite amount of time,
what are the axioms of the object language.
If it didn't work in your case, that's unfortunate.
--
Daryl McCullough
Ithaca, NY
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