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

On 6/29/2026 12:05 PM, André G. Isaak wrote:
> On 2026-06-29 07:29, 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. The way that terms-of-the-art are formed
>> is very misleading and as much as intentionally deceptive.
> 
> You really need to learn that terms used in a given field have 
> definitions within that field that may or may not correspond to what you 
> want a term to mean, and that you actually need to learn those definitions.
> 

That is the way that it usually works. In Proof Theoretic
Semantics each author has their own terms-of-the-art that
has a very similar yet not exactly the same semantic meaning
as entirely different terms-of-the-art used by another author.

Also these meanings gradually evolve over time so they
change in subtle ways from their original meanings.

> In mathematics, a system is incomplete if there are statements in the 
> language of that system which can neither be proven nor disproven.
> 

Q was intentionally defined to handle less than PA
thus is not at all in any way incomplete relative
to its defined purpose.

> That's *all* incomplete means. No more, no less. It doesn't mean that 
> something is missing that could be added. It makes no reference 
> whatsoever to the purpose for which a system was designed.
> 

So they could have defined "has a box of clowns" as
the situation where en expression can neither be
proven nor refuted in Q.

So we can say that the halting problem "has a box
of clowns" instead of saying that computation is
in any way limited.

> When mathematicians talk about rings, do you object based on the fact 
> that you can't put them on your finger?
> 
> When mathematicians talk about fields, do you object based on the fact 
> that nothing can graze on them?
> 
> To put things in terms of your system, the term 'incomplete' as used by 
> mathematicians has a different GUID than the term 'incomplete' when used 
> colloquially, just as the term 'pen' has different GUIDs depending on 
> whether it is used to store pigs or ink. [note that I do not actually 
> endorse the use of GUIDs; that's just plain silly].
> 
> André
> 

And likewise "undecidable" really means that the
expression is semantically incoherent. We could
equally call this "has a square box of clowns".


-- 
Copyright 2026 Olcott

My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.

The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.

My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.

(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).