Date: Feb 7, 2013 6:51 PM
Author: fom
Subject: Re: Matheology § 210
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.