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

On 2026-06-29 15:39, olcott wrote:
> On 6/29/2026 4:18 PM, André G. Isaak wrote:
>> On 2026-06-29 15:10, olcott wrote:
>>> On 6/29/2026 3:58 PM, André G. Isaak wrote:
>>>> On 2026-06-29 14:06, olcott wrote:
>>>>> On 6/29/2026 3:02 PM, André G. Isaak wrote:
>>>>>> On 2026-06-29 13:47, olcott wrote:
>>>>>>> On 6/29/2026 2:33 PM, André G. Isaak wrote:
>>>>>>>> On 2026-06-29 13:08, olcott wrote:
>>>>>>>>> On 6/29/2026 1:29 PM, André G. Isaak wrote:
>>>>>>>>
>>>>>>>>>> Is "has a box of clowns" in the language of Q? No. I didn't 
>>>>>>>>>> think so, so your example is completely irrelevant.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> It is an idiom stipulated to mean:
>>>>>>>>> sentences in the language of Q which can neither
>>>>>>>>> be proven nor disproven by Q
>>>>>>>>
>>>>>>>> Q doesn't have idioms. That's a natural language concept alien 
>>>>>>>> to theories of arithmetic.
>>>>>>>>
>>>>>>>>>>> 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?
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Off topic, irrelevant.
>>>>>>>>>
>>>>>>>>>>>> When mathematicians talk about fields, do you object based 
>>>>>>>>>>>> on the fact that nothing can graze on them?
>>>>>>>>>>
>>>>>>>>>> No answer?
>>>>>>>>
>>>>>>>> These questions are Irrelevant because
>>>>>>> In Proof Theoretic Semantics
>>>>>>> statements in the language of that system
>>>>>>> which can neither be proven nor disproven
>>>>>>>
>>>>>>> have not established that they have semantic
>>>>>>> meaning because semantic meaning is ONLY
>>>>>>> established in PTS by canonical proofs.
>>>>>>
>>>>>> This is a misrepresentation on your part. Whereas truth functional 
>>>>>> semantics takes true and false to be the semantic primatives, PTS 
>>>>>> uses either (depending on which author you follow) proven and not 
>>>>>> proven or provable and not provable as its primitives without 
>>>>>> dealing with truth or falsity. 
>>>>>
>>>>> Yes that is an accurate paraphrase.
>>>>>
>>>>>> Thus, they would treat a statement like 'no number is greater than 
>>>>>> its successor' as being unprovable in Robinson Arithmetic, not as 
>>>>>> being meaningless as you seem to think.
>>>>>>
>>>>>
>>>>> You are not being consistent with you own paraphrase.
>>>>> I still don't have all of the exact nuances exactly
>>>>> correct because unlike every other field each author
>>>>> has their own terms-of-the-art.
>>>>
>>>> Of course I am being consistent. Within PTD, unproven/unprovable 
>>>> *is* a semantic value, 
>>>
>>> Impossibly provable in Q means cannot possibly
>>> derive a semantic meaning Q.
>>
>> 'impossibly' in English is an intensifier, i.e. 'he was impossibly 
>> strong' means 'he was exceedingly strong'. I have no idea what 
>> 'impossibly provable' might mean, but if you intended to say 
>> 'unprovable' then you are misinterpreting PTS. Unprovable is one of 
>> the two semantic primitives used by PTS (the other being provable).
>>
> 
> This exactly and perfectly what it precisely means.

If it means 'unprovable' then say 'unprovable' or 'impossible to prove'. 
Don't use a nonsensical expression like 'impossibly provable'.

> % This sentence is not true.
> ?- LP = not(true(LP)).
> LP = not(true(LP)).
> ?- unify_with_occurs_check(LP, not(true(LP))).
> false.

That's an example, not a definition. Examples don't take the place of 
definitions.

>>>> i.e. a meaning; so you can't claim that the expression 'no number is 
>>>> greater than its successor' isn't meaningful in Q.
>>>>
>>>> Can you provide a single example of someone working within PTS who 
>>>> has taken issue with incompleteness? Incompleteness exists in PTS 
>>>> just as much as it exists in any other framework.
>>
>> I would really like you to answer the above question.
> 
> If you understand PTS you will understand that their
> reasoning cannot possibly get to incompleteness.

Then you should be able to produce an actual citation to this effect. As 
it stands, this is simply a baseless assertion on your part, and since 
your grasp of PTS doesn't seem particularly strong, it carries very 
little weight.

You keep offering PTS as an alternative to truth-functional semantics, 
but incompleteness has absolutely nothing to do with truth-functional 
semantics as the definition of incompleteness doesn't even mention truth 
or falsity (or semantics). If anything, it is more aligned with PTS than 
with TFS since it pertains to theoremhood i.e. provability, the semantic 
primitive used by PTS.

A system is incomplete if there is an expression in the language of that 
system, P such that neither P nor ¬P can be derived as a theorem. 
'Theorem' is a notion pertaining to provability, not truth.

André

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