On 2/7/2013 10:42 AM, WM wrote: > On 7 Feb., 15:49, fom <fomJ...@nyms.net> wrote: >> On 2/7/2013 7:54 AM, WM wrote: >> >>> Once upon a time there used to be a logocal identity: The expression >>> "Set X is countable" used to be equivalent to "Set X can be listed". >> >> Incorrect. > > First learn, then understand, then judge. > >
I did. I took the time to tolerate your nonsense. During that time I learned how your behavior is motivated by an agenda rather than any honest and sincere respect for mathematical investigations. Understanding that (with a little help from others) I judged it was time to keep your nonsense honest.
> Otherwise you appear silly.
How one appears to a WMoron is irrelevant.
> > 1.1 Definition The cardinality of A is less than or equal to the > cardinality of B (notation: |A| ? |B|) if there is a one-to-one > mapping of A into B.
Coming from you, a true definition of any kind is a breath of fresh air -- except that this definition has nothing to do with the logical priority of well-order over cardinality, or, for that matter, your inappropriate use of the word "list" and its derivatives in a context where it has no commonly understood use.
But, since you so conveniently chose Jech, note that his definition of the class of alephs relies on the notion of a least ordinal for any given class of sets that cannot be distinguished on the basis of your definition.
The problem with what you tried to do, going back to basics, can be found in Aristotle. One can never prove a definition. But one can destroy a definition. This is the epistemic nature of whichever equivalence underlies any definition. Although Cantor, himself, never quite recognized the logical priority of well-ordering in full, his definition of "cardinal number" is not the definition of "cardinality" you gave. That general definition is meaningless beyond a classificatory role without a canonical representative to which a determinate sense of "number" is attached. This is implicit in Cantor's observation that the objects of a set are abstracted to "units" as part of the abstraction from a set to its "cardinal number". Jech's version of set theory does not abstract to "units". It fixes specific ordinal numbers as the alephs.