In a deposition submitted under oath, Relatif Tuinn said:
 ND> That's right. Hmm. Your number is still infinity, is it not?
 RT> Yes, but it is a bigger one than you stated. Here is a bigger one than
 RT> Mark's: infinity^infinity
    If you want to compete with infinite numbers, then I've got you both
 beat with aleph-one.  Before I can give a more formal definition, allow
 me to give some background.  My apologies if you know this already; I
 don't mean to insult anyone's intelligence.
    Transfinite numbers, also called ordinal numbers, are typically
 described as some number n as an ordered set M such that if you could
 count M in the correct order, then you could count up to n.  So
 transfinite numbers arise through counting, and the are two principles
 for generating ordinal numbers: 1) if you have the ordinal number n,
 then you can find a next ordinal n + 1; 2) if you have some definite
 sequence of increasing ordinals n, then you can find a last ordinal
 which is greater than all the n's, called lim(n), or the limit of n.
    Starting with 0 (zero is the first ordinal after the empty
 sequence), the first principle can be repeatedly applied to get the
 ordinal numbers 0, 1, 2, ....  To get past the infinite sequence of
 finite ordinals, we use principle 2 to get lim(n), usually called w
 (which is as close as I can come to a lower-case Greek omega).  Omega
 is sometimes called aleph-null.  So, now we have 0, 1, 2, ..., w, and
 can continue with w + 1, w + 2, using principle 1, and principle 2
 applied here gives us lim(w + n) which is the same as w + w, or w * 2.
    These are called countable infinities, because they have a
 one-to-one relationship with Z+, the set of positive integers.  This
 relationship also means that w, w + w, w^2 and w^w are the same number,
 because there is a one-to-one map from w + w to w.  Aleph-one is the
 first ordinal that cannot be mapped one-to-one onto w.  One way of
 describing aleph-one is this: consider the set of all polynomial
 functions in x with natural number coefficients (e.g., x, x + 3, x^2 +
 4x, x^3, etc.) ordered such that the graph of one is steeper than the
 previous.   If S is the set of functions such that for every g(x) in S
 there will always be an f(x) in S that is steeper, then S must have at
 least aleph-one members.
    Of course, there is also an aleph-two which is bigger, and an even
 larger aleph-three, on up to and past aleph-aleph.  The Greek symbol of
 a capital omega supposedly represents the absolute infinity that lies
 beyond all ordinals, but it is pretty much inconceivable.  As if
 aleph-one *was* conceivable... ;^)  Big Omega, the Absolute Infinity,
 is the ultimate "my infinity is bigger than your infinity" game.  And
 I, therefore, have The trump card!  I win!  I win!  (grin)
... Parturiunt montes, nascetur ridiculus mus.
--- PPoint 2.05
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