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

On 04/07/2026 16:31, Tristan Wibberley wrote:
> On 06/05/2026 20:37, Julio Di Egidio wrote:
>> On 02/05/2026 20:47, Scott Hoge wrote:
>>
>>> In Cantor's theorem, we do not actually construct a diagonal.
>>> Rather, we presuppose that we can enumerate a set, and then,
>>> /purely on the grounds of possibility/, conceive a diagonalized
>>> non-element.
>>
>> Nope, as explained and re-explained ad nauseam around here:
>> just the resident trolls won't get it.
>>
>> Cantor's diagonal argument, the one with the binary sequences,
>> is indeed constructive: a definition of anti-diagonal of *any*
>> (infinite) list is provided, and the proof that the anti-diagonal
>> cannot be in the list is quite constructive.
> 
> "quite" but not "completely".
> 
> A constructive operation is defined, but a diagonal number is
> constructed just when that constructive operation is applied to a
> constructible list. 

I should note for the less knowledgable readers of course it's less
often than that, it is only that often for systems such as the one Julio
and Phoenix are using which allows dequantification of universally
quantified statements into the system proper which then have derivable
statements containing actual constructions of the constructible objects
they apply to by virtue of their original quantification. Of course,
dequantification of fantastically quantified statements doesn't make a
statement about nonconstructible objects because there aren't any
outside of the fantastical quantification.

By which I don't mean to argue the countability of the set of reals as
defined in what we call Cantor's Proof of the Uncountability of the
Reals to include objects quantified over by fantatstical quantification
but not by universal quantification, but it does make some meaning clearer.

While some of the sets might have objects in the system proper, some of
the members of some of the sets clearly do not.

-- 
Tristan Wibberley

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