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

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. 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.

André


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