In a deposition submitted under oath, William Elliot said:
 BS> infinity + infinity = infinity = infinity ^ 2 = infinity ^ infinity,
 WE> No!!!  2^infinity > infinity.  A diagonal argument can prove this.
 WE> Are you familiar with set theory?
    I am familiar with set theory, yes.  I am not familiar with what a
 diagonal argument is, unless you are referring to the Cantor
 diagonalization process.
 WE> Otherwise a digital proof can be made for denumerable (countable)
 WE> infinity.  Namely that there are -more- real numbers than rational
 WE> numbers.  Hence a -larger- non-denumerable or uncountable infinity
 WE> in addition to the smaller denumerable or countable infinity.
    Infinities are strange.  Yes, you can have infinite sets of
 different sizes, but that is not always what is implied when I say
 w + w = w (where I use w to be the lowercase omega as a symbol for any
 particular but arbitrarily chosen infinite set) or w ^ 2 = w.  Using
 the same symbol implies the same set.
    "Big deal" you might say, "that changes nothing about what I said."
 Well, it does.  You would be right if w was some cardinal number.
 w has a cardinality, but w is a concept, not a real number, and the
 rules change a bit when dealing with infinities, or at least are not as
 intuitive as they are when dealing with finite numbers.
    Yes, the set of real numbers is larger than the set of rational
 numbers, but I can show you that the set of positive rational numbers
 is equal to the set of all rational numbers which is equal to the set
 of all positive integers which is equal to the set of all integers.
    The best way I know to clarify my point is to show some proofs.  To
 do this, we must clarify some definitions and show some basic proofs.
    I assume you are familiar with the following math definitions, but
 for the benefit of any other interested parties reading along, I'll
 give them here, in as informal a way as possible.  Bear with me,
 because understanding these definitions, basic theorems and initial
 proof are necessary to understanding my ultimate point, coming in a
 later message.
    One-to-one (OTO): a function is OTO if, and only if, every x plugged
 into the function results in a unique y.  IOW, if f(x1) = f(x2) then
 x1 = x2.  The function x^3 is an example of a function which is OTO.
 Every x in the domain leads to a unique y in the range of the function.
 There may be some y's without corresponding x's, but no x's without
 corresponding y's.
    ONTO: a function is ONTO if, and only if, every element y in the
 range of the function has an x in the domain that takes the function to
 that y.  IOW, for every y, there exists an x such that y = f(x).  There
 may be some x's without corresponding y's, and some x's may lead to the
 same y, but no y's without corresponding x's.
    One-to-one correspondence:a function that is both OTO and ONTO.
 Every x maps to a unique y and visa versa; no x's are left without
 a y and visa versa.
    Now, more into sets.  Let A and B be any sets.  A has the same
 cardinality as B if, and only if, there is a one-to-one correspondence
 from A to B, i.e., there is a function from A to B that is OTO and
 ONTO.
    Theorem:  For all sets A, B and C,  (i) A has the same cardinality
 as A (reflexive property of cardinality), (ii) if A has the same
 cardinality as B, then B has the same cardinality as A (the symmetric
 property of cardinality) and (iii) if A has the same cardinality as B,
 and B has the same cardinality as C, then A has the same cardinality as
 C (the transitive property of cardinality).
    A set is countably infinite if, and only if, it has the same
 cardinality as the set of positive integers Z+.  A set is countable if,
 and only if, it is finite or countable infinite.  A set that is not
 countable is called uncountable.
    One must understand the above to make sense of the rest of my
 proofs, except maybe this first one, which was shown in my previous
 post in a different manner than this:
    Continued next message...
... I am logged on, therefore I am.
--- PPoint 2.05
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