We know that the real numbers of set theory are very different from the real numbers of analysis, at least most of them, because we cannot use them. But it seems, that also the natural numbers of analysis 1, 2, 3, ... are different from the cardinal numbers 1, 2, 3, ...
This is a result of the story of Tristram Shandy, mentioned briefly in § 077 already, who, according to Fraenkel and Levy ["Abstract Set Theory" (1976), p. 30] "writes his autobiography so pedantically that the description of each day takes him a year. If he is mortal he can never terminate; but if he lived forever then no part of his biography would remain unwritten, for to each day of his life a year devoted to that day's description would correspond."
This result is counter-intuitive, but set theory needs the feature of completeness for the enumeration of all rational numbers. If not all could be enumerated, the same cardinality of |Q and |N could not be proved.
Nevertheless, matheologians deny every contradiction. One of them, Michael Greinecker (as a self-proclaimed watchdog, and bouncer in MathOverflow http://meta.mathoverflow.net/discussion/1296/crank-post-to-flag-as-spam/#Item_0 an interbreeding of Tomás de Torquemada and Lawrenti Beria) stated: "there is no contradiction. Just a somewhat surprising result. And there is no a apriory reason why one should be able to plug in cardinal numbers in arithmetic formulas for real numbers and get a sensible result." This means the finite positive integers differ significantly from the finite positive cardinals or, as Cantor called them, the finite positive integers. Well, maybe, sometimes evolution yields strange results. But if they differ, how can set theory any longer be considered to be the basis of analysis?