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Re: Matheology § 224
Posted:
Apr 9, 2013 5:06 AM


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 Allmightyformalsystem, proving all arithmetic >>> true formulas. >>> >>> We'd be simply change the name "Allmightyformalsystem" >>> to "Allmightylanguagestructure". But it's still an Incompleteness >>> (of the 2nd kind) that we'd encounter: the Incompleteness of language >>> structure interpretation of the abstract (nonlogical) 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 trueable, or falseable, 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 firstorder 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



