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

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*

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

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

> 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. 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

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