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

On 2026-06-29 12:16, olcott wrote:
> On 6/29/2026 12:05 PM, André G. Isaak wrote:
>> 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.
>>
> 
> That is the way that it usually works. In Proof Theoretic
> Semantics each author has their own terms-of-the-art that
> has a very similar yet not exactly the same semantic meaning
> as entirely different terms-of-the-art used by another author.
> 
> Also these meanings gradually evolve over time so they
> change in subtle ways from their original meanings.
> 
>> In mathematics, a system is incomplete if there are statements in the 
>> language of that system which can neither be proven nor disproven.
>>
> 
> Q was intentionally defined to handle less than PA
> thus is not at all in any way incomplete relative
> to its defined purpose.

The definition of 'incomplete' makes no reference whatsoever to 'defined 
purpose'. If there are sentences in the language of Q which can neither 
be proven nor disproven by Q, then Q is incomplete. And it is.

>> 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.
>>
> 
> So they could have defined "has a box of clowns" as
> the situation where en expression can neither be
> proven nor refuted in Q.

Is "has a box of clowns" in the language of Q? No. I didn't think so, so 
your example is completely irrelevant.

> So we can say that the halting problem "has a box
> of clowns" instead of saying that computation is
> in any way limited.
> 
>> When mathematicians talk about rings, do you object based on the fact 
>> that you can't put them on your finger?

No answer?

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

No answer?

André

>> 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
>>
> 
> And likewise "undecidable" really means that the
> expression is semantically incoherent. We could
> equally call this "has a square box of clowns".
> 
> 

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