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

On 2026-06-29 07:29, olcott wrote:
> On 6/29/2026 1:14 AM, Mikko wrote:
>> On 29/06/2026 05:52, olcott wrote:
>>> On 6/28/2026 3:39 AM, Mikko wrote:
>>>> On 27/06/2026 17:50, polcott wrote:
>>>>> On 6/27/2026 1:53 AM, Tristan Wibberley wrote:
>>>>>> On 20/06/2026 18:32, olcott wrote:
>>>>>>
>>>>>>> A proof theoretic expression is known to be true when
>>>>>>> it is fully grounded in its atomic base. Only two
>>>>>>> PTS semantics researchers deal with true Dag Prawitz
>>>>>>> is the one that began this. PTS previously only dealt
>>>>>>> with semantic meaning and never got around to true(L,x).
>>>>>>
>>>>>> That's surprising, disregard for axioms?
>>>>>
>>>>> If there is no sequence of inference steps in Q from
>>>>> ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) is
>>>>> ungrounded in the PTS atomic base of Q.
>>>>>
>>>>> This does not mean undecidable or incomplete
>>>>> it means that ~∃x x=S(x) is out-of-scope for Q.
>>>>
>>>> It comes close. If ∃x x=S(x) is likewise "ungrounded" but in the
>>>> language of Q then ~∃x x=S(x) and ∃x x=S(x) are both undecidable
>>>> and Q is incomplete, bcause that is what the words mean.
>>>
>>> Q also can't bake a birthday cake, this does not make
>>> Q in any way "incomplete" relative to what it was
>>> defined to do. Incomplete only counts relative to
>>> its intended purpose. A car without an engine is
>>> incomplete relative to a mode of transportation.
>>
>> Irrelevant. The definition of completeness 
> 
> It a misnomer and does not literally mean (as it implies)
> that something is missing that could be added to make
> it complete. The way that terms-of-the-art are formed
> is very misleading and as much as intentionally deceptive.

You really need to learn that terms used in a given field have 
definitions within that field that may or may not correspond to what you 
want a term to mean, and that you actually need to learn those definitions.

In mathematics, a system is incomplete if there are statements in the 
language of that system which can neither be proven nor disproven.

That's *all* incomplete means. No more, no less. It doesn't mean that 
something is missing that could be added. It makes no reference 
whatsoever to the purpose for which a system was designed.

When mathematicians talk about rings, do you object based on the fact 
that you can't put them on your finger?

When mathematicians talk about fields, do you object based on the fact 
that nothing can graze on them?

To put things in terms of your system, the term 'incomplete' as used by 
mathematicians has a different GUID than the term 'incomplete' when used 
colloquially, just as the term 'pen' has different GUIDs depending on 
whether it is used to store pigs or ink. [note that I do not actually 
endorse the use of GUIDs; that's just plain silly].

André

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