> 0.999(9) + 0.999(9)
> = 1.999(9)8

Where did you get the 8 from? There is a 9 after EVERY 9, so each 9 + 9 gets
a carry from the two 9's after them. To say that there is a last 9 (at
inifinity) is denying the 'unendiness'. If such a reasoning (an identifiable
'last 9') were accepted al the beautifull reasonings about infinities (e.g.
Cantors diagonal argument) would be invalid.

I guess most of you know the funny aspects of unendiness (of the first kind,
aleph-0):

A hotel has infinite rooms, all of which are occupied. One more guest
arrives, and he gets room 0. The guest in room 0 moves to room 1, 1 moves to
2, etcettera. Now all the old guests + the one new guest have a room.
[ aleph0 + 1 = aleph0 ]
[ aleph0 + n = aleph0 for each n in N ]

Two hotels A and B, each with infinite rooms, all occupied, merge. Hotel A
is sold. The guest from rooms A(n) (for all n) move to the rooms B(2*n). The
guest in rooms Bn move to room B(2*n+1). All guest from both hotels now have
a room in hotel B.
[ 2 * aleph0 = aleph0 ]
[ n * aleph0 = aleph0 for each n in N ]

As an excercise for the reader: show that infinite hotels, each with
infinite rooms, can merge without loss of capacity. Note that it is not
enough to repeat the previous argument ad infinitum, because aleph0 is not a
member of N.
[ aleph0 * aleph0 = aleph0 ]

But there are limits. Cantors' diagonal argument shows that the number of
rational numbers is larger than aleph-0.

I really loved this subject way back at university. Sadly they also teached
me a lot of less interesting subjects, like algebra....

Wouter

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