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

On 06/07/2026 18:47, olcott wrote:
> On 7/6/2026 4:17 AM, Mikko wrote:
>> 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.
> 
> It fully meets its design spec thus calling it
> any kind of incomplete is a damned lie.

The definitions of "complete" and "complete" don't refer to desing
specs.

But, because you meantioned it, we would want to know what is the
design spec of Q or where can we find it.

-- 
Mikko