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

On 7/4/2026 1:07 PM, olcott wrote:
> On 7/4/2026 3:06 AM, Mikko wrote:
>> 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.
>>
> 
> The English word "incomplete" establishes the base
> meaning (parent node) in the knowledge ontology.
> 
> I will not tolerate deceptive terms-of-the-art.

In other words, you intend to lie by misusing definitions.

> 
> A motor vehicle that lacks a motor is incomplete.
> Q that cannot resolve (∀x, S(x) ≠ x) is complete
> according to its definition.
>