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

On 30/06/2026 16:58, olcott wrote:
> On 6/30/2026 3:18 AM, Mikko wrote:
>> 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. 
> 
> Base-Extension Semantics (B-eS) allows that.
> It never was incomplete. It always did what it was defined to do.
> When Q is extended to become PA it stops being Q and becomes PA.

However, there are theories that reamain incomplete even when
more postolates are added, as long as there is a way to know
which sentences are included in the added postulates. Important
examples include Peano arithmetic and ZFC set theory.

-- 
Mikko