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

On 6/29/2026 6:45 PM, André G. Isaak wrote:
> 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'.
> 

It only means that: when formalized.

> 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
> 

Because they beat around the bush about that using
terminology that varies across every author.

>> "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.
>>
>>
> 


-- 
Copyright 2026 Olcott

My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.

The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.

My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.

(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).