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

On 6/29/2026 5:25 PM, André G. Isaak wrote:
> On 2026-06-29 15:39, olcott wrote:
>> On 6/29/2026 4:18 PM, André G. Isaak wrote:
>>> On 2026-06-29 15:10, olcott wrote:
>>>> On 6/29/2026 3:58 PM, André G. Isaak wrote:
>>>>> On 2026-06-29 14:06, olcott wrote:
>>>>>> On 6/29/2026 3:02 PM, André G. Isaak wrote:
>>>>>>> On 2026-06-29 13:47, olcott wrote:
>>>>>>>> On 6/29/2026 2:33 PM, André G. Isaak wrote:
>>>>>>>>> On 2026-06-29 13:08, olcott wrote:
>>>>>>>>>> On 6/29/2026 1:29 PM, André G. Isaak wrote:
>>>>>>>>>
>>>>>>>>>>> Is "has a box of clowns" in the language of Q? No. I didn't 
>>>>>>>>>>> think so, so your example is completely irrelevant.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> It is an idiom stipulated to mean:
>>>>>>>>>> sentences in the language of Q which can neither
>>>>>>>>>> be proven nor disproven by Q
>>>>>>>>>
>>>>>>>>> Q doesn't have idioms. That's a natural language concept alien 
>>>>>>>>> to theories of arithmetic.
>>>>>>>>>
>>>>>>>>>>>> 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?
>>>>>>>>>>>
>>>>>>>>>>> No answer?
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Off topic, irrelevant.
>>>>>>>>>>
>>>>>>>>>>>>> When mathematicians talk about fields, do you object based 
>>>>>>>>>>>>> on the fact that nothing can graze on them?
>>>>>>>>>>>
>>>>>>>>>>> No answer?
>>>>>>>>>
>>>>>>>>> These questions are Irrelevant because
>>>>>>>> In Proof Theoretic Semantics
>>>>>>>> statements in the language of that system
>>>>>>>> which can neither be proven nor disproven
>>>>>>>>
>>>>>>>> have not established that they have semantic
>>>>>>>> meaning because semantic meaning is ONLY
>>>>>>>> established in PTS by canonical proofs.
>>>>>>>
>>>>>>> This is a misrepresentation on your part. Whereas truth 
>>>>>>> functional semantics takes true and false to be the semantic 
>>>>>>> primatives, PTS uses either (depending on which author you 
>>>>>>> follow) proven and not proven or provable and not provable as its 
>>>>>>> primitives without dealing with truth or falsity. 
>>>>>>
>>>>>> Yes that is an accurate paraphrase.
>>>>>>
>>>>>>> Thus, they would treat a statement like 'no number is greater 
>>>>>>> than its successor' as being unprovable in Robinson Arithmetic, 
>>>>>>> not as being meaningless as you seem to think.
>>>>>>>
>>>>>>
>>>>>> You are not being consistent with you own paraphrase.
>>>>>> I still don't have all of the exact nuances exactly
>>>>>> correct because unlike every other field each author
>>>>>> has their own terms-of-the-art.
>>>>>
>>>>> Of course I am being consistent. Within PTD, unproven/unprovable 
>>>>> *is* a semantic value, 
>>>>
>>>> Impossibly provable in Q means cannot possibly
>>>> derive a semantic meaning Q.
>>>
>>> 'impossibly' in English is an intensifier, i.e. 'he was impossibly 
>>> strong' means 'he was exceedingly strong'. I have no idea what 
>>> 'impossibly provable' might mean, but if you intended to say 
>>> 'unprovable' then you are misinterpreting PTS. Unprovable is one of 
>>> the two semantic primitives used by PTS (the other being provable).
>>>
>>
>> This exactly and perfectly what it precisely means.
> 
> If it means 'unprovable' then say 'unprovable' or 'impossible to prove'. 
> Don't use a nonsensical expression like 'impossibly provable'.
> 

Impossibly provable because remains stuck
in an infinite loop.

>> % This sentence is not true.
>> ?- LP = not(true(LP)).
>> LP = not(true(LP)).
>> ?- unify_with_occurs_check(LP, not(true(LP))).
>> false.
> 
> That's an example, not a definition. Examples don't take the place of 
> definitions.
> 

It is the only perfect example of an idea from
Proof Theoretic Semantics that seems to stay a
little bit nebulous because each author uses their
own author specific terminology.

>>>>> i.e. a meaning; so you can't claim that the expression 'no number 
>>>>> is greater than its successor' isn't meaningful in Q.
>>>>>
>>>>> Can you provide a single example of someone working within PTS who 
>>>>> has taken issue with incompleteness? Incompleteness exists in PTS 
>>>>> just as much as it exists in any other framework.
>>>
>>> I would really like you to answer the above question.
>>
>> If you understand PTS you will understand that their
>> reasoning cannot possibly get to incompleteness.
> 
> Then you should be able to produce an actual citation to this effect.

Each author uses their own author specific terminology
and the meanings slightly change across authors.

>  As 
> it stands, this is simply a baseless assertion on your part, and since 
> your grasp of PTS doesn't seem particularly strong, it carries very 
> little weight.
> 
> You keep offering PTS as an alternative to truth-functional semantics, 
> but incompleteness has absolutely nothing to do with truth-functional 
> semantics as the definition of incompleteness doesn't even mention truth 

Truth as an Epistemic Notion --- Dag Prawitz
What is the appropriate notion of truth for
sentences whose meanings are understood in
epistemic terms such as proof or ground for
an assertion? It seems that the truth of such
sentences has to be identified with the existence
of proofs or grounds...
https://link.springer.com/article/10.1007/s11245-011-9107-6

> or falsity (or semantics). If anything, it is more aligned with PTS than 
> with TFS since it pertains to theoremhood i.e. provability, the semantic 
> primitive used by PTS.
> 
> A system is incomplete if there is an expression in the language of that 
> system, P such that neither P nor ¬P can be derived as a theorem. 

That is simply not the way that it works in Proof Theoretic Semantics.

> 'Theorem' is a notion pertaining to provability, not truth.
> 
> André
> 


-- 
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).