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

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, 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.

André

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