🚀 go-pugleaf

"Greg d. Moore" <mooregr@greenms.com> wrote previously:
*}All Number set:   1 1.1 1.11 1.111 1.1111 1.11111 ...... up to 2.
*}Integers	  1  2   3    4     5      6   up to infinity.
*}So, (wrap your mind around this :-) you get to infinity with the
*}integers before you get to 2.  So what happens when you finally get to
*}2?  You're out of inifite integers, but you've got an inifity to go. :-)

Although it is indeed easy to show with the Cantor diagnalization that
the real numbers outnumber the integers, the above method is
disasterously bad as an effort at a proof.

Hint:  Two sets are the same size if they can be put in 1-to-1 (onto)
correspondence.  The positive integers and all integerts can easily be
mapped in such a manner as given earlier.  However, merely trying the
failed correspondence such as:

Positive Integers   1, 2, 3, 4, . . .
All Integers       -1,-2,-3,-4, . . .

does NOT show that after you attempt correspondence all the positive
integers are "LEFT OVER" from the count of all integers.  All you've
shown is that you picked the wrong order to list the two sets.

Similarly, all Moore has done is show that he can find a wrong order to
list the reals and integers.  The existence of such wrong orders is
trivial.  The goal is to show that there IS NO RIGHT ORDER.  See
Cantor's elegant proof of this.

Yours, Lulu...

quilty@  _/_/_/_/ THIS MESSAGE WAS BROUGHT TO YOU BY: \_\_\_\_    n o
philos. _/_/             Postmodern Enterprises            \_\_
umass. _/_/                                                 \_\_  d o
edu   _/_/_/ IN A WORLD W/O WALLS, THERE WOULD BE NO GATES \_\_\_ z e