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.