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

On 7/6/2026 1:54 PM, André G. Isaak wrote:
> On 2026-07-06 12:12, olcott wrote:
>> On 7/6/2026 12:56 PM, André G. Isaak wrote:
>>> On 2026-07-06 11:45, olcott wrote:
>>>> 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.
>>>
>>> Yes, I agree that it is a lie.
>>>
>>> For starters, there's no such thing as the 'natural preexisting order 
>>> of the body of knowlege'.
>>>
>>
>> Sure there is. There is a minimal sized knowledge ontology.
>> Anything less than minimal wastes RAM and CPU cycles.
> 
> Saying something is preexisting means it has always been around; before 
> there was RAM or CPU cycles; before there were people to know things.
> 

Mathematical incompleteness does have a proper
place in the knowledge ontology that does not
inherit from incomplete(base)

unfulfilled_goals seems to be a more accurate base
for mathematical incomplete. Q does not do what we
want it to do even though it was intentionally defined
to only be a fragment of PA some people still want
it to do what PA does.

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.

> A specific computer model might implement a knowledge ontology, but 
> that's hardly 'preexisting'. And if it implements something where the 
> mathematical meaning of 'incomplete' inherits from some other definition 
> of 'incomplete' then that model does not correspond to reality.
> 
> André
> 
>>> And incomplete(math) doesn't inherit from anything so claiming it 
>>> inherits from incomplete(base) (whatever that may be) is of course a 
>>> lie.
>>>
>>> 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).