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

On May 24, 12:48 pm, 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.

I retorted:
That is completely incoherent.

AK continued:
I'm sure it is to you.

Which was indefensibly abusive, as well as being
utter bull***. I was NOT expressing an opinion,
***. It is incoherent TO EVERYbody, INCLUDING
the people who profess to believe it, INCLUDING YOU.
The fact that non-Abelian groups exist simply has nothing
to do with anything being "a false theorem".
THERE IS NO SUCH THING as a false theorem.

Curiously, most people studying logic are perfectly
capable of appreciating the mathematical theorem

Meta-mathematical theorem.

that consistent theories can prove false statements,

CALLING that a "theorem" is almost as idiotic as calling
the statements in question "false". WE ARE IN A *CONTEXT*
here. THIS context is DIFFERENT from the one in which
the observation you just cited gets to call ITSELF a "theorem".
WE are in a context where the word "theorem"
implies A THEORY, which is CLOSED under
LOGICAL CONSEQUENCE. Structures and interpretations
generally are NOT terribly relevant to that because the
THEORY, the statements IN the set closed under consequence,
is ALL TRUE in ALL MODELS of ITSELF!
More to the point, it is true under all models of THE AXIOMS
FROM WHICH it was closed, if it was recursively axiomatizable.
INside the theory (where any statement that gets inside is
inside because IT IS a THEOREM), no such thing AS A FALSE
*statement*, let alone "false theorem", occurs. False statements
occur
UNDER INTERPRETATIONS, IN STRUCTURES, or (just barely)
IN MODELS, and in the latter case, they are ALL NOT proved and
therefore NOT THEOREMS!
But even when you have an unproved statement that you are calling
false in one model of the axioms, that very statement, unless it,
in addition to being unproved, IS ALSO DISproved, will be TRUE
in some OTHER model of the SAME axioms! So even CALLING
it "a false statement" withOUT THE QUALIFICATION of the model
under which it came up false IS NOT DEFENSIBLE. In the case
of things like PA, the qualification is implicit because everybody
always knows which model you mean. IN THE CASE OF MOST
OTHER THINGS, THIS IS NOT the case. In the case where
the statement is disproved, it is false, but again, it is not both
false
AND A PROVEN THEOREM, and more to the point, it is not MERELY
false: it is false not ONLY in whatever model you might've been
using to justify resort to terms as weak as "true" and "false", it
is false IN EVERY, ANY, AND ALL models of the axioms AS WELL!
To call such a statement MERELY "false" is lying by
omission&understatement.

The fact that I am attacking a common linguistic practice
does NOT imply that I don't understand something, except
maybe why anybody as well-educated as AK will call you
stupid (after having talked with you about the subject for
over a decade) for doing it.



in the sense that while they might initially
find it puzzling, after explanations and examples
they understand the mathematical content of the assertion,
and can follow the proof.

To understand the mathematical content of the assertion
and to follow the proof means to understand that the
conclusion is true in all models of the axioms, if the thing
is in fact proved, as it in fact must be, IN ORDER to be
A THEOREM. A theorem is the conclusion OF A PROOF.
It is therefore true IN ALL models of the axioms from which
the conclusion was proved. Therefore, it is not merely
oxymoronic, but moronic, to say "false theorem".

But you already knew I thought all that, so none of that is the
point. The point is that you think (or at least *claim*; I'm
going to give you credit for NOT being stupid enough to actually
THINK this; I'm going to counter-claim that this is comes out of
some sort of tragic EMOTIONAL animus on your part) that my belaboring-
this-point-past-the-point-of-its-being-PAINFULLY-obvious
shows that you can prove some metatheorem that I don't understand.
That is obviously not the case. I OBVIOUSLY (this has been
obvious to you FOR A LONG time, which makes what you have
said above COMPLETELY indefensible) know that when you SAY
"false" about a statement of this type in this context, YOU MEAN
"false in some subclass of structures for which we think we can defend
some sort of privilege". Since you know I know that, what power or
principality is persuading you to abuse me by falsely claiming
publicly
that I do not know it? C O U L D I T B E ....
................................ SATAN^H^H^H^H^Hthe ghost
of Torkel Franzen??????????????????


I liked your characterisation of my style as "arrant pedantry" better,

That wasn't global. That was an example of you being arrantly
pedantic at that time. This time you're just lying abusively.


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