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

On 03/07/2026 21:20, olcott wrote:
> On 7/3/2026 12:35 PM, André G. Isaak wrote:
>> On 2026-07-03 09:38, olcott wrote:
>>> On 7/3/2026 4:28 AM, Mikko wrote:
>>>> On 02/07/2026 17:49, olcott wrote:
>>>>> On 7/2/2026 1:55 AM, Mikko wrote:
>>>>>> 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.
>>>>>
>>>>> Is  (∀x, S(x) ≠ x) provable or refutable in Q?
>>>>> Yes if you cheat, no if you don't cheat.
>>>>
>>>> As I already pointed out in another message, which you apparently
>>>> missed, it is neither. 
>>>
>>> Thus PTS would say that (∀x, S(x) ≠ x) is semantically
>>> undefined in Q.
>>
>> And that differs from claiming that Q is incomplete exactly how...?
> 
> The base definition of "incomplete" means that it is
> not operating according to design spec.

No, it is not. The term "incomplete" in its base meaning is
appicable to various things that are not exprected to operate.

-- 
Mikko