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

On 04/07/2026 20:07, 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.
> 
> A motor vehicle that lacks a motor is incomplete.

It is so incomplete that it is not a motor vehicle until a motor
is installed.

A motor vechicle that lacks brakes and head lights is a motor
vehicle but incomplere and, depending on the place and time,
may be unacceptable for public roads. Installing the head lights
makes it more complete but it is still incomplere as long as
no breaks are installed.

> Q that cannot resolve (∀x, S(x) ≠ x) is complete
> according to its definition.

By the defintion of "incomplete" Q is incomplete. The theory
Q + (∀x, S(x) ≠ x) is more complete but still incomplete.
This is perfectly analogous to the motorvehicle without head
lights and brakes: the meaning of "incomplete" is the same
although the definition is different.

-- 
Mikko