Date: Apr 9, 2013 5:06 AM
Author: Alan Smaill
Subject: Re: Matheology § 224
Nam Nguyen <namducnguyen@shaw.ca> writes:

> On 08/04/2013 8:30 AM, Alan Smaill wrote:

>> Nam Nguyen <namducnguyen@shaw.ca> writes:

>>

>>> Seriously, we should begin to abandon the idea that whatever is true

>>> or false in the naturals can be structure theoretically proven,

>>> verified.

>>>

>>> 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

> place.

What reasoning steps are allowed in showing properties of

language structures?

> 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.

--

Alan Smaill