Re: Godel's incompleteness and formal language

From: Truth Detector (TD_at_evaluator.com)
Date: 02/02/05 

Date: Wed, 02 Feb 2005 21:12:52 GMT

Jim Spriggs <jim.sprigs@ANTISPAMbtinternet.com.invalid> wrote in
news:42013628.2C6D94FE@ANTISPAMbtinternet.com.invalid:

> Truth Detector wrote:
>>
>> First, is "axiomatizable system" equivalent to "formal language?"
>
> No, as well as a language there is a deducibility relation.

That isn't included? First Order logic is an undecidable formal language
that includes that relation. Is there a formal language that doesn't have
such a relation?
>
>>
>> If so, what is the appropriate way to state Godel's theorem in terms
>> of formal languages.
>>
>> I know this isn't right:
>>
>> No formal language is able to express all truthful statements.
>>
>> They can be expressed, they just can't all be determined (T or F). In
>> fact The theorem explicitly states that there are "statements"
>> expressible in that language that cannot be decided.
>>
>> So it looks like it should be:
>>
>> No formal language can demonstrate that all truthful statements
>> expressible in that language are in fact true.
>>
>> Except that seems kind of circular, so here's another try:
>>
>> All formal languages have statements that are true, but cannot be
>> shown to be true.
>>
>> Unfortunately, in this form, I can't see why formal languages have
>> this limitation. They clearly do! But I'm trying to see what it is
>> that limits them.
>
> Two important things: it's all formal systems that contain a certain
> amount of arithmetic.

Yes, I forgot to include that. But it's not just arithmetic, but the
equivalent to something more than '+' and '*'. First order logic doesn't
use arithmetic, per se, but does have equivelents.

>
> And it's "... are true but cannot be _proved_."
>
I don't know how you could show something to be true without proving it.

TD

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