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

On 01/07/2026 18:16, olcott wrote:
> On 7/1/2026 2:24 AM, Mikko wrote:
>> 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.
> 
> Base-Extension Semantics (B-eS) seems to be essentially a cheat.
> When we ask what is grounded in an atomic base of Q and we
> add axioms to Q to become PA we cheated in that we changed
> the original question rather than answered it.

Yes, in a sense. But sometimes it is better to have a partial answer
rather than no answer at all. Of course Q with any additional
postulate is not Q but if the additional postulates are true about
natural numbers then the strengthened theory is still a theory of
natural numbers. PA is one such strengthened Q but still incomplete
and can be strengthened further.

-- 
Mikko