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Topic: Matheology § 224
Replies: 6   Last Post: Apr 11, 2013 12:51 AM

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Alan Smaill

Posts: 770
Registered: 1/29/05
Re: Matheology § 224
Posted: Apr 9, 2013 5:06 AM
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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



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