Dear Tim,
You are reiterating what students still have to learn, and there is almost no interest in a clarification because the idea of cardinality is obviously of no use, and generations of mathematicians failed to disprove Cantor's proof. Among the listed and also the not listed opponents were excellent minds like Aristotle, Galileo, Leibniz, Newton, Cauchy, and Poincaré. Cantor was certainly arrogant when he declared them wrong altogether. They understood that an infinite quantity or an infinite series is by definition of infinite as the opposite of finite an unfinished process, not something that has been completed via counting. If we speak of "all" natural numbers then this means excluding the possibility to add a larger number, not because already all natural or rational numbers are already occupied but because, as formulated by Archimedes and the axiom of infinity, the property of being infinite means being open to unlimited enlargement.
When Cantor and Fraenkel postulated a fixed series of "all" natural numbers, they appealed on thinking in terms of counting discrete elements and they implicitly denied that being infinite is not at all a quantity but a quality. Cantor fell back into primitive mistakes by Albert of Saxony (1316-1390) and Bernhard Bolzano (1781-1848) who attributed points to a space or a Menge (a set) to a line, respectively. Cantor managed to humiliate Kronecker because the latter also intended but failed to make the continuum rigorously algebraic. It was already and is still undisputed that the expression infinity must not always be algebraically treated. For instance, it is impossible to increase or decrease infinity by addition, subtraction and other operations. Likewise the evidence "2^aleph_0 is larger than aleph_0" by Cantor/Hessenberg is based on treating infinity like a number. Mediocre mathematicians were and are perhaps still unable to think beyond the mathematical formalisms they learned. Therefore they could not even disprove naïve set theory.
Already Galileo used bijection in order to show that there are not more natural numbers than squares of it because both series are endless which implies they are uncountable in the original sense. This is logically convincing to me.
Cantor confused the world with uncommon definitions. He defined the natural and rational numbers countable in the sense the latter can be put into 1 to 1 correspondence with the natural numbers. Obviously, his Mächtigkeit (cardinality) of countable (according to Cantor's definition) infinity is nothing else than the property of being discrete and therefore numerically distinguishable. This is the logical opposite of the property of being continuous.
What about the axiom of extensionality, I see the same problem as with Dedekind's smaller than, equal to, or larger than relation for the continuum of real numbers (elements of measure zero).
I didn't intend insulting any proponent of Cantor's naïve set theory when I mentioned its mysticism. The attribute naïve stems from those who tried to circumvent the logical inconsistencies of Cantor's set theory and make it seemingly less mystic while even less concrete.
In order to get an impression how Cantor impressed the experts, one may read how lecturing he reacted to justified question. Emmy Nöther reported how Cantor theatrical answered the question how he imagined infinity as follows: He directed his view towards the sky, his eyes starred to infinity, after a while he performed a slow wide movement with his hand, and spoke with pathos: I see it an abyss. The more insane he got, the more he was considered a genie.
Don't forget: Point Set Theory didn't lead to anything of value although Bertrand Russell meant: "The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast." No comment.
Regards,
Eckard
