Re: what makes it true?

On 7 Sep 2005 17:50:57 GMT, rusin@xxxxxxxxxxxxxxxxxxxxx (Dave Rusin)
wrote:

>In article <dfmb23$77o$1@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
> <mareg@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:
>
>>But my understanding of GC is that
>>it is a statement about the natural numbers rather than
>> a formal statement in the theory of the natural numbers.
>
>What does this mean? I don't know what a "statement about but not
>a formal statement in" would even refer to, but isn't
>
> forall n exists p,q forall r,s
> 2n = p+q & (p=r*s ==> (p=r or p=s)) & (q=r*s ==> (q=r or q=s))
>
>a formal statement in the theory of the natural numbers?

Yes, it is. Well, formally, the theory of the natural numbers
is the collection of all statements in a certain language
which are _true_ of the natural numbers, so nobody knows
whether this is a statement in that theory. What you meant
was to say that it's a formal statement in the _language_
of the natural numbers...

What's above might have been better expressed

"But my understanding of GC is that
it is a statement about the natural numbers rather than
statement about some formal theory."

Saying that GC is true is the same as saying that
GC is an element of the theory of the natural numbers,
but one might, at least informally, say that the
theory of the natural numbers is not a "formal"
theory. One could say that if one were taking
"formal theory" to mean "the set of consequences
of some (recursive) set of axioms" - we know that
nobody knows a list of axioms that characterize
the theory of the natural numbers.

(re "recursive" above: Of course if we simply take
every statement true in the natural numbers to be an
axiom then we have a list of axioms that completely
characterize this theory. But that doesn't count,
because if we do that then there's no algorithmic test
to determine whether or not a given statement is an
axiom. Axioms are kind of silly if you can't tell
what's an axiom and what's not...)

>dave


************************

David C. Ullrich
.

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