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

On 29/06/2026 16:29, olcott wrote:
> On 6/29/2026 1:14 AM, Mikko wrote:
>> On 29/06/2026 05:52, olcott wrote:
>>> On 6/28/2026 3:39 AM, Mikko wrote:
>>>> On 27/06/2026 17:50, polcott wrote:
>>>>> On 6/27/2026 1:53 AM, Tristan Wibberley wrote:
>>>>>> On 20/06/2026 18:32, olcott wrote:
>>>>>>
>>>>>>> A proof theoretic expression is known to be true when
>>>>>>> it is fully grounded in its atomic base. Only two
>>>>>>> PTS semantics researchers deal with true Dag Prawitz
>>>>>>> is the one that began this. PTS previously only dealt
>>>>>>> with semantic meaning and never got around to true(L,x).
>>>>>>
>>>>>> That's surprising, disregard for axioms?
>>>>>
>>>>> If there is no sequence of inference steps in Q from
>>>>> ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) is
>>>>> ungrounded in the PTS atomic base of Q.
>>>>>
>>>>> This does not mean undecidable or incomplete
>>>>> it means that ~∃x x=S(x) is out-of-scope for Q.
>>>>
>>>> It comes close. If ∃x x=S(x) is likewise "ungrounded" but in the
>>>> language of Q then ~∃x x=S(x) and ∃x x=S(x) are both undecidable
>>>> and Q is incomplete, bcause that is what the words mean.
>>>
>>> Q also can't bake a birthday cake, this does not make
>>> Q in any way "incomplete" relative to what it was
>>> defined to do. Incomplete only counts relative to
>>> its intended purpose. A car without an engine is
>>> incomplete relative to a mode of transportation.
>>
>> Irrelevant. The definition of completeness 
> 
> It a misnomer and does not literally mean (as it implies)
> that something is missing that could be added to make
> it complete.

It does mean that something is missing that could be added to
enabe a proof of an unprovable sentence. For example, the theory
of a group does not tell whether ∀x∀y (x * y = y * x). It could
be true or it could be false. But it is possible to add an
axiom to make it provagle, e.g, x∀y (x * y = y * x) itself,
giving the theory of an Abelian group, which is still incomplete
but can be said to be less incomplete. Therefore the meaning of
"complete" is not very different from its common sense meaning,
just not as ambiguous.

-- 
Mikko