Re: Cantor's circular "proof" that evens = integers

On Jun 3, 9:59 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:

As to formalisation,

There are a lot more Americans than Britons reading this.
Would it really kill you to spell that with a z?
I'm not just being tendentious here. The question of whether
American English and British English really are DIFFERENT
LANGUAGES is actually going to get raised by this argument.

pointing out that there are models of the language of
arithmetic -- that's cue for another of your standard rants --

Precisely because that cues a standard rant, it follows that
*I*HAVE*NEVER* pointed THAT out! NEVER!
I DON'T ACKNOWLEDGE THE EXISTENCE of a
language of arithmetic, so *I* could *never* point THAT out!

In the first place, neither you nor anyone else acknowledges the
existence of MODELS of the language of arithmetic. YOU mean
L-STRUCTURES into which the language of arithmetic may be
INTERPRETED, or INTERPRETATIONS of the language.

*I*, *BY* *CONTRAST*, mean MODELS of the AXIOMS of PA
(not the CORRECT use of MODELS there), so to be analogous
with ME, you are going to HAVE to use MY terms and MY
paradigm! IF you had had enough integrity to do that, you would've
AVOIDED falling headlong into the ditch into which you fall below.

which Prov_PA('A') for some sentence
A comes out true even if A is not provable in
PA is as pertinent an observation,
as far as the question of whether Prov_PA
formalises provability in PA,

***. If all you are doing is interpreting the language,
then your interpretation might not only give true as a truth-
value for a false assertion of provability, it might also give
true as a truth-value for A COUNTEREXAMPLE *TO*AN*AXIOM*
of PA! The relevant point for the problem we are facing here
IS NOT, as you both pig-headedly and maliciously imply,
that some bizarre interpretation could yield a bizarre result.
It is that many interpretations CAREFULLY CONSTRAINED
TO MATCH GOOD AXIOMS *still* fall short.

as suggesting that the fact that "omenat kasvavat
puussa" (which is Finnish for "apples grow on trees") is false if we,
for some bizarre reason, count carrots as 'omena's,
and is thus not an accurate
translation of "apples grow on trees".

There are no bizarre reasons going on here.
If you ever do decide to count carrots as 'omena's, that
will be because of some revolutionary approach to plant
genetics or some discovery that we've been misreading both
genomes all these years. Otherwise that will simply be prohibited
BY THE THEORY and BY THE AXIOMS governing both the language
and the science. The LANGUAGE is NOT relevant and is NOT primary.
THE AXIOMS are primary. Since the axioms have to be stated
IN a language, you still need a dictionary, both in PA AND
in the botany departments of Finnish universities.


Provability enters into questions about formalisation

Oh, shut up.
You cannot defend your order-of-mention of these two terms.
ALL PROOF IS INHERENTLY formal. Therefore ALL questions
of BOTH provability and formalization
START INHERENTLY INTERTWINED ALL the time.
NOBODY EVER BOTHERS "formalizing" ANYthing EXCEPT
to facilitate proof!


only when we need ensure the formalisation -- depending e.g. on
this or that coding of sequences of naturals -- is equivalent to other
canonical formalisations, over some weak base theory, for proof theoretic
purposes.

NOW who is being abstruse to the point of queerness -- AS OPPOSED
to simplistic??


.

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