Date: Apr 9, 2013 5:06 AM
Author: Alan Smaill
Subject: Re: Matheology § 224
Nam Nguyen <firstname.lastname@example.org> writes:
> On 08/04/2013 8:30 AM, Alan Smaill wrote:
>> Nam Nguyen <email@example.com> writes:
>>> Seriously, we should begin to abandon the idea that whatever is true
>>> or false in the naturals can be structure theoretically proven,
>>> If we don't, we'd be in _no_ better position than where Hilbert
>>> was with his All-mighty-formal-system, proving all arithmetic
>>> true formulas.
>>> We'd be simply change the name "All-mighty-formal-system"
>>> to "All-mighty-language-structure". But it's still an Incompleteness
>>> (of the 2nd kind) that we'd encounter: the Incompleteness of language
>>> structure interpretation of the abstract (non-logical) concept known
>>> as the natural numbers.
>> But it is known structure theoretically that if we have any 2 structures
>> that satisfy Peano axioms, then they are isomorphic: a statement
>> is true in one if and only if it's true in the other.
> Provided that the statement is true-able, or false-able, in the first
What reasoning steps are allowed in showing properties of
> And cGC isn't that kind of statement.
You already said IIRC that you can express cGC using the language
of first-order Peano arithmetic. That means that if cGC is true
in *any* one language structure where the PA axioms are true, then
it's true in *every* such structure; it cannot be true in one
such structure and false in another.