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

On 7/6/2026 12:27 PM, André G. Isaak wrote:
> On 2026-07-06 10:58, olcott wrote:
>> On 7/6/2026 11:07 AM, André G. Isaak wrote:
>>> On 2026-07-06 09:47, olcott wrote:
>>>> On 7/6/2026 4:17 AM, Mikko wrote:
>>>>> On 04/07/2026 20:07, olcott wrote:
>>>
>>>>>> Q that cannot resolve (∀x, S(x) ≠ x) is complete
>>>>>> according to its definition.
>>>>>
>>>>> By the defintion of "incomplete" Q is incomplete. The theory
>>>>> Q + (∀x, S(x) ≠ x) is more complete but still incomplete.
>>>>
>>>> It fully meets its design spec thus calling it
>>>> any kind of incomplete is a damned lie.
>>>
>>> What exactly do you think the 'design spec' of Q is? 
>>
>> Make sure that Q has less capability than PA is its design
>> spec by its designer.
> 
> And you presumably have a reference to back that up?
> 

It is common knowledge that was Robinson's purpose

In mathematics, Robinson arithmetic is a finitely
axiomatized fragment of first-order Peano arithmetic
(PA), first set out by Raphael M. Robinson in 1950.
It is usually denoted Q.

https://en.wikipedia.org/wiki/Robinson_arithmetic

> But it doesn't matter either way since the mathematical definition of 
> incomplete makes no reference to the 'spec' of a system.
> 

Within the natural preexisting 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.

Math could have as accurately specified the
word-label "F has a squirrel in its socks"
as its meaning of math incomplete.

> A system is incomplete if there exists some statement P such that 
> neither P nor ¬P can be derived as theorems of that system.
> 
> Importantly, this definition doesn't 'inherit' anything from any other 
> definition of 'incomplete' which might exist. That's not how language 
> actually works.
> 
> 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).