Re: The simple essence of Proof Theoretic Semantics

On 2026-07-03 12:12, olcott wrote:
> On 7/3/2026 12:17 PM, André G. Isaak wrote:
>> On 2026-07-03 10:48, olcott wrote:
>>> On 7/3/2026 9:45 AM, André G. Isaak wrote:
>>>> On 2026-07-02 23:02, olcott wrote:
>>>>> On 7/1/2026 9:03 PM, olcott wrote:
>>>>>> Q cannot do the ∀x without an infinite sequence of steps.
>>>>>
>>>>> So your phrasing is good: Q would need something like an infinite 
>>>>> sequence of steps (or a single principle that summarizes them) to 
>>>>> get the ∀x. Since formal proofs must be finite, and Q lacks the 
>>>>> tool (induction) that would allow a finite proof of the infinite 
>>>>> claim, the universal statement remains unprovable.
>>>>
>>>> I'm not sure why you are responding to yourself nor who 'your 
>>>> phrasing' refers to since you don't quote anyone. But, assuming 
>>>> we're still talking about ∀ x, S(x) ≠ x in Q, your reasoning is 
>>>> simply off.
>>>>
>>>> You *can* prove universally quantified claims in Q, just not that 
>>>> particular claim.
>>>>
>>>
>>> What is the reason that (∀x, S(x) ≠ x) cannot be proved in Q?
>>
>> Because it isn't true in all models of Q, 
> Model theory has been expressly off-topic for
> many weeks in every thread. Whenever you ignore
> this the rest of your reply will be ignored.

The rest of my post which you snipped and (presumably) ignored explained 
*why* you are wrong about this. PTS does not reject models or model 
theory. It simply doesn't rely on model-theoretic semantics. Q 
*requires* a model.

André

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