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

On May 24, 9:50 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
It is also important to note that that a consistent theory might prove false
statements is a mathematical observation no more problematic than, say, the
existence of non-Archimedean fields or non-Abelian groups.

That is completely incoherent.
Nobody claims that non-Archimedean fields are NOT fields
or that non-Abelian groups are NOT groups.
But non-standard models of PA really do have things
in them that are NOT natural numbers.

"that a consistent theory might prove false statements"
SIMPLY NEVER HAPPENS.
YOU cannot even make sense of it.
Those of US who make sense of it know that making
sense of it REQUIRES PRIVILEGING an interpretation
in which an axiom IS FALSE, which is COMPLETELY
perverse: THE WHOLE PURPOSE of defining axioms is
TO FOCUS attention on interpreations where the axioms
ARE TRUE! Every *model* is by definition a model OF
some AXIOMS!

In order to EVEN ASCRIBE a truth-value to anything,
YOU MUST RETREAT *from* the theory to SOME PROPER
subset of its interpretations. Things that are proved by the theory
are true, and things that are disproved by it are false, IN ALL
the relevant interpretations! So NOTHING EVER "proves a false
theorem" and in the case of things that are proved or disproved,
they ARE MORE than merely true and MORE than merely false:
they are MODALLY, NECESSARILY true or false, IF YOU
TAKE YOUR AXIOMS *as* *AXIOMS*!!

Which, of course, you have NO CHOICE BUT to do --
"axioms ARE axioms" IS analytic, for crying out loud.

AK is about to get very abusive regarding what the community
does, and I am about to get equally contemptuous reactively
about how the community could stand to do a FEW things
DIFFERENTLY.


.

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