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

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.

> 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).