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

On 2026-06-29 17:36, olcott wrote:
> On 6/29/2026 6:03 PM, André G. Isaak wrote:
>> On 2026-06-29 16:38, olcott wrote:
>>> On 6/29/2026 5:25 PM, André G. Isaak wrote:
>>>> 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'.
>>>>
>>>
>>> Impossibly provable because remains stuck
>>> in an infinite loop.
>>
>> You're abusing English. As I said, 'impossibly' is an intensifier. If 
>> I say someone is impossibly strong it doesn't mean it is impossible 
>> for them to be strong, it means they are stronger than I would have 
>> thought possible, i.e. that they are extraordinarily strong. Saying 
>> something is 'impossibly provable' would mean it is extraordinarily 
>> provable which isn't coherent since provability isn't a gradient 
>> concept. What is wrong with simply using the term 'unprovable' which 
>> is actually coherent English?
>>
> 
> 3 is impossibly numerically greater than 5.

Not if you're speaking English.

>>>>> % 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.
>>>>
>>>
>>> It is the only perfect example of an idea from
>>> Proof Theoretic Semantics that seems to stay a
>>> little bit nebulous because each author uses their
>>> own author specific terminology.
>>
>> It has absolutely nothing to do with Robinson Arithmetic or 
>> incompleteness which were the topics under discussion. 
> 
> It perfectly establishes that impossibly provable
> means has no proof theoretic semantic meaning.

It shows no such thing.

> There is a key difference between we did not yet
> find a proof of X and a proof of X cannot possibly
> exist.

Yes. And the two English terms in use for these are 'unproven' and 
'unprovable'. Not 'impossibly provable'.

>> It's your feeble attempt at trying to formalize the liar paradox in 
>> Prolog and it fails at that because the Liar Paradox rests on the 
>> interpretation of the deictic expression 'this', and your formulation 
>> does not contain anything corresponding to 'this'. It is simply a 
>> circular definition.
>>
> 
> "this" literally means := when formalized
> LP := ~True(LP) expands to
> ~True(~True(~True(~True(~True(~True(~True(...)))))))

No. := means defined as, not 'this'.

If you think otherwise, explain how you would formalize a sentence 
involving what you would call non-pathological self reference using :=. 
For example, 'this sentence contains five words'.

>>>>>>>> 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.
>>>
>>> Each author uses their own author specific terminology
>>> and the meanings slightly change across authors.
>>
>> How does this prevent you from offering a citation?
>>
> 
> It does seem that they do agree that no proof
> of G can possibly exist in PA does means that
> G has no semantic meaning in PA.

I've not seen anyone operating in PTS who says anything remotely like 
that. They do not agree with this; rather, you are projecting your own 
peculiar views onto their theory.

André

> "Failure of Normalization"
> "Lack of a Canonical Form"
> "Disharmony"
> are some of the ways that they describe this.
> 
> I have to spend a very long time carefully
> analyzing two papers before I can even use
> one author's terms regarding one aspect of PTS
> limited to that author's terms.
> 
> "anti-realism" seems like it means a psychotic
> break for reality yet seems to merely specify
> valid deductive inference in the terms-of-the-art
> of PTS.
> 
> 

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