In a deposition submitted under oath, William Elliot said:
 BS> Infinities are strange.  Yes, you can have infinite sets of
 BS> different sizes, but that is not always what is implied when I say
 BS> w + w = w (where I use w to be the lowercase omega as a symbol for
 BS> any particular but arbitrarily chosen infinite set) or w ^ 2 = w.
 BS> Using the same symbol implies the same set.
 WE> This is questionable notation. as lower case omega is used for the
 WE> ordinal number of the integers in ascending order.  I suggest capital
 WE> letters infinity such as A, B for aleph, beth as in A0 aleph null or
 WE> denumerable infinity.  It is true that for A or B infinite, A + B = A
 WE> * B = max(A,B) while for A  B
    I saw this notation in a book _Infinity and the Mind_ by Rudy
 Rucker.  I adopted it here because I can't find the infinity key on my
 keyboard.  ;^)   But I'll go with A(0) for aleph-null, etc.
 BS> One-to-one (OTO): a function is OTO if, and only if, every x
 BS> plugged into the function results in a unique y.  IOW, if f(x1) =
 BS> f(x2) then x1 = x2.  The function x^3 is an example of a function
 BS> which is OTO. Every x in the domain leads to a unique y in the range
 BS> of the function. There may be some y's without corresponding x's, but
 BS> no x's without corresponding y's.
 WE> A 1-1 function is an invertible function, a bijection.
    Well, try injective.  A bijection is a 1-1 correspondence.
 BS> ONTO: a function is ONTO if, and only if, every element y in the
 BS> range of the function has an x in the domain that takes the function
 BS> to that y.  IOW, for every y, there exists an x such that y = f(x).
 BS> There may be some x's without corresponding y's, and some x's may lead
 BS> to the same y, but no y's without corresponding x's.
 WE> This is a weak definition.
    Yes, perhaps because I wanted to make this clear to anyone not
 familiar with mathematical notation and proofs, hence tried to simplify
 it.  Had I wanted to be as precise as possible, I would have defined
 a function F:X->Y as ONTO (or surjective) iff for every y in the set Y,
 there exists an x in the set X | F(x) = y.
 BS> A set is countably infinite if, and only if, it has the same
 BS> cardinality as the set of positive integers Z+.  A set is countable
 BS> if, and only if, it is finite or countable infinite.  A set that is
 BS> not countable is called uncountable.
 WE> I use the word denumerable to mean countably infinite.
    Okay.  I'll try and remember that.
 BS> One must understand the above to make sense of the rest of my proofs,
 WE> Learned this stuff while still in high school.
    I've just been learning it the last year or so in college.
... And smale foweles maken melodye that slepen al the nyght with open eye.
--- PPoint 2.05
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