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

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Posts: 2,777
Registered: 12/13/04
Re: Matheology § 224
Posted: Apr 9, 2013 9:21 PM
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On 09/04/2013 3:06 AM, Alan Smaill wrote:
> 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?

A structure theoretical property is just an n-ary predicate set
of (n-tuples). Hence, the step of showing the property is quite simple:
to structure theoretically verify that, per a given formula F, a certain
set of n-tuples _is a subset of the underlying predicate set_ .
(Note a function set is also a predicate set).

If we don't know how to verify whether or not a set is a subset of
another set, it's very much a forgone conclusion we simply can't
argue anything about language structure: about the natural numbers,
about Goldbach Conjecture, about cGC, etc... .

>> 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.

If the set of ordered pairs you perceived as a language structure
can be used to verify the truth of cGC at all.

Remember, a language structure is something you must create per the
FOL definition of language structure. And until you _verify_ what
you've created is indeed a language structure, some bets will be off
(so to speak).

So, can you verify that what you perceive as the natural numbers
be indeed a language structure for L(PA)?

There is no remainder in the mathematics of infinity.


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