In a deposition submitted under oath, William Elliot said:
 >>> Bob Sewell on Infinity
 BS> How can you have something bigger than something that never ends? I
 BS> don't actually believe it; I got most of this from a book called
 BS> Infinity and the Mind by Rudy Rucker.
 WE> The integers and the real numbers.  The integers never end 1,2,3,....
 WE> but the reals never end either 1,2,3... but between every two are
 WE> infinity many more. between 1 and 2, 2 and 3, etc.  But between every
 WE> two numbers in the infinity of numbers between 1 and 2 is yet another
 WE> infinity and so on without limit infinitely more and more numbers
 WE> cramming in between infinities of numbers. These infinities within
 WE> infinities are so dense that you can't list them one after another
 WE> like you can integers.  What's the 'next' number after 1/2?  .51,
 WE> .501, .5001, ...?  See, you can't tell me the next.
    Yes, I've seen the proof for this.  It shows that the cardinality of
 the real numbers between any integer n and its neighbor integer n + 1
 (or n - 1) are uncountably infinite.  Understanding this proof on a
 rational, logical level doesn't make it any easier for my mind to
 comprehend an infinity larger than another on the level of common
 sense.  Because I can never empirically perceive *any* infinity, I
 can't empirically perceive one infinity larger than another.
    It seems to be the same problem many people have with the idea of
 God.  They cannot or will not believe in what they cannot empirically
 perceive.
 BS> But, Mr Rucker says that, unlike finite ordinals, infinite
 BS> ordinals are not commutative, therefore 1 + A = A, but A + 1 = A + 1,
 BS> and 2 * A = A, but A * 2 = A + A.  Furthermore, starting with 0 and
 BS> continuously adding 1, you count through the ordinals to get:
 WE> Be careful about notation as w is the ordinal number of the integers,
 WE> A0, the cardinal.
    Sorry.  All this contemplation of infinities makes my brain hurt.
 It's a small wonder Cantor went insane.
 BS> 0, 1, 2, ..., A, A + 1, A + 2, ..., A * 2, A * 2 + 1, A * 2 + 2, ...,
 BS> A * 2 + A (aka A * 3), and continue through A * n for each finite n,
 BS> and on to A * A, which is also A ^ 2, then to A ^ 3, A ^ 3, to A ^ A
 BS> and on even further.
 WE> w^w, w^w^w, etc until w^w^w^w^.....  Drives you crazy, polynomials of
 WE> denumerable ordinals.  Finite ordinals and cardinals are the same, so
 WE> that's why this distinction is so novel.
    And difficult to understand.
 BS> So, I guess you are right and I was mistaken.  But, like I said, I
 BS> really don't believe there is any number past A because the very
 BS> definition of A seems to me to include any and all numbers greater
 BS> than A.
 WE> Let's assume that all of the real numbers in the unit interval are
 WE> denumerable. So we can list all of them.  Let a1 a2 etc be digits as
 WE> also b1, c1 and so on for as many digits as we need.  Note each real
 WE> number in the unit interval is a denumerable series of digits.  So I
 WE> list the numbers:
 WE> #1 = 0.a1 a2 a3 a4 ...
 WE> #2 = 0.b1 b2 b3 b4 ...
 WE> #3 = 0.c1 c2 c3 c4 ...
 WE> #4 = 0.d1 d2 d3 d4 ...
 WE> etc for 5,6,7...
 WE> Now I show there is a real number #x not in this list.
 WE> ...
 WE> This is the diagonal argument.
    Yes, this is the proof to which I referred above.  However, my
 Discrete Math textbook calls the proof of the countability of the set
 of positive rational numbers which I showed in the last post to you
 Cantor's diagonal proof.
 WE> Are you familiar with set theory?
    Yes.
 WE> Do you know about P(S) the power set of S which the set of all sets
 WE> included in the set S, {x | x included in S}?  A simple generalized
 WE> diagonal argument can be used to show the the cardinality of P(S) is
 WE> greater than the cardinality of S itself. So we can continue S, P(S),
 WE> P(P(S)), etc with a never ending series of ever larger infinities. 
 WE> But that's not the end, the whole series can be leap frogged to an
 WE> even greater number than all of the previous numbers.  There's no end
 WE> to this series of ever greater infinities.
    I know.  It still doesn't make it easier on a common sense level.
 And it makes my brain hurt to try to make sense of it on that level.
... Certe, Toto, sentio nos in Kansate non iam adesse.
--- PPoint 2.05
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