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

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 simply the same way that actual
meaning really works. Model theory is merely
a strange idea that was adopted because keeping
semantics and syntax together as one was too
difficult for people to figure out back when
model theory was first created.

-- 
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).