Date: Feb 3, 2013 4:29 PM
Author: fom
Subject: Re: Matheology § 208

On 2/3/2013 2:23 AM, WM wrote:
> Again to quote a passage from Das Kontinuum:
>

?inexhaustibility? is essential to the
> infinite.
>


This is the particular reason for properly
distinguishing between "transfinite" and
"infinite".

One can certainly reject transfinite arithmetic
as mathematics. But, Cantor did develop a
calculus of arithmetical operations that one
can call a system. Ultimately, however, he
was confronted with Kant's definition

"Infinity is plurality without unity"

which may perhaps be from an earlier writer.

It is, however, interesting to consider
Aristotle with respect to "counting" infinity,
its "inexhaustibility," and the techniques of
modern logic:

"It would be absurd if we had principles
innately; for then we would possess knowledge
that is more exact than demonstration, but
without noticing it [...]

"Clearly, then, we must come to know the
first things by induction; for that is also
how perception produces the universal in us."

So, one has a choice between Kant's a priori
synthetic knowledge (which was developed to
counter Hume's skepticism) or Aristotle. The
modern paradigm rejects Kant and conducts
its foundational research accordingly.

Hierarchies of increasingly strengthened
theories reflect the inexhaustibility
of infinity and truth in the classical
sense is replaced by truth persistence
(under forcing for classical set-theoretic
foundations).

As for Kant, George Boolos showed a
bit of honesty when he wrote:

"We need not read any contemporary theories
of the a priori into the debate between Frege
and Kant. But Frege can be thought to have
carried the day against Kant only if it has
been shown that Hume's principle is analytic,
or a truth of logic. This has not been
done. [...]

"Well. Neither Frege nor Dedekind showed
arithmetic to be a part of logic. Nor did
Russell. Nor did Zermelo or von Neumann.
Nor did the author of Tractatus 6.02 or his
follower Church. They merely shed light
on it."

Perhaps that is why Frege retracted his
logicism at the end of his career.