Re: Within Proof Theoretic Semantics Gödel's G h as no meaning in PA

On 2026-07-06 10:58, olcott wrote:
> On 7/6/2026 11:07 AM, André G. Isaak wrote:
>> On 2026-07-06 09:47, olcott wrote:
>>> On 7/6/2026 4:17 AM, Mikko wrote:
>>>> On 04/07/2026 20:07, olcott wrote:
>>
>>>>> Q that cannot resolve (∀x, S(x) ≠ x) is complete
>>>>> according to its definition.
>>>>
>>>> By the defintion of "incomplete" Q is incomplete. The theory
>>>> Q + (∀x, S(x) ≠ x) is more complete but still incomplete.
>>>
>>> It fully meets its design spec thus calling it
>>> any kind of incomplete is a damned lie.
>>
>> What exactly do you think the 'design spec' of Q is? 
> 
> Make sure that Q has less capability than PA is its design
> spec by its designer.

And you presumably have a reference to back that up?

But it doesn't matter either way since the mathematical definition of 
incomplete makes no reference to the 'spec' of a system.

A system is incomplete if there exists some statement P such that 
neither P nor ¬P can be derived as theorems of that system.

Importantly, this definition doesn't 'inherit' anything from any other 
definition of 'incomplete' which might exist. That's not how language 
actually works.

André

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