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

On 06/07/2026 20:11, olcott wrote:
> On 7/6/2026 4:25 AM, Mikko wrote:
>> On 05/07/2026 00:01, olcott wrote:
>>> On 7/4/2026 12:11 PM, dbush wrote:
>>>> 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.
>>>
>>> These definitions are the liars.
>>
>> Maybe your definitions, but not the usual ones, which tell truthfully
>> how the defined words are used and understood by the experts.
> 
> Within the natural preexisting order of the body
> of knowledge

There is no natural pre-existing order of the body of knowledge.

> saying that incomplete(math) inherits
> part of its meaning from incomplete(base) semantic
> parent node is simply a lie.

Then don't say so.

> Using the code word of "cat" for a dalmatian dog
> is equally dishonest.

Then don't use that word, or if you need to, add "sensu ..." where
... is the author whose definition you refer to.

> It violates the natural
> preexisting order of the body of knowledge.

The natural order of the body of knowledge is not pre-existing,
it is non-existing.

>>> Term-of-the-art
>>> A cat is a dalmatian dog

>> Perhaps some art but neither ailurology nor cynology.
-- 
Mikko