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

On 7/6/2026 3:08 PM, André G. Isaak wrote:
> On 2026-07-06 13:03, olcott wrote:
>> 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)
> 
> It doesn't inherit from *anything*
> 


The preexisting order of all knowledge is constructed
incrementally on the basis of the root of {thing}.

> A system is incomplete if there exists some statement P such that 
> neither P nor ¬P can be derived as theorems of that system.
> 

What would be its parent node?

> That's what it means. Nothing more. Nothing less. It doesn't acquire any 
> aspect of its meaning from anything else.
> 

It is impossible to leap from {nothingness} to
1987 Chevy Camaro with no steps inbetweem.

{Thing}--->{Physically Existing Thing}
  ... {Motor Vehicle}---> {Automobile} ...

>> 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.
> 
> It doesn't *have* a base. 

That you do not understand how ideas are derived from
other ideas is less than no rebuttal at all.

> It simply means what it means (or it is its 
> own base if you want to look at it like that). The definition makes no 
> mention whatsoever of what me may *want* a system to do.
> 
>> 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.
> 
> Yes. I know what Robinson Arithmetic is. There's really no reason for 
> you to explain it.
> 
> 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).