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.