>>> Bob Sewell on Infinity 
 BS> I am familiar with set theory, yes.  I am not familiar with what a
 BS> diagonal argument is, unless you are referring to the Cantor
 BS> diagonalization process.
That I am.
 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.
This is questionable notation. as lower case omega is used for the ordinal 
number of the integers in ascending order.  I suggest capital letters 
infinity such as A, B for aleph, beth as in A0 aleph null or denumerable 
infinity.  It is true that for A or B infinite, A + B = A * B = max(A,B) 
while for A  B
 BS> Yes, the set of real numbers is larger than the set of rational
 BS> numbers, but I can show you that the set of positive rational numbers
 BS> is equal to the set of all rational numbers which is equal to the set
 BS> of all positive integers which is equal to the set of all integers.
Simple enuf.
 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.
A 1-1 function is an invertible function, a bijection.
 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.
This is a weak definition.  A function f is onto a set S if the range of f is 
s.  Note that the range of f is the map of the domain.  If the range of f is 
strictly included in the set S then the function f is considered into s.
 BS> One-to-one correspondence:a function that is both OTO and ONTO.
 BS> Every x maps to a unique y and visa versa; no x's are left without
 BS> a y and visa versa.
A 1-1 correspondence between two sets S and T, is a bijection from S onto T.
 BS> Now, more into sets.  Let A and B be any sets.  A has the same
 BS> cardinality as B if, and only if, there is a one-to-one
 BS> correspondence from A to B, i.e., there is a function from A to B that
 BS> is OTO and ONTO.
S and T are equinumerous if there a bijection from S to T.
 BS> Theorem:  For all sets A, B and C,  (i) A has the same cardinality
 BS> as A (reflexive property of cardinality), (ii) if A has the same
 BS> cardinality as B, then B has the same cardinality as A (the symmetric
 BS> property of cardinality) and (iii) if A has the same cardinality as
 BS> B, and B has the same cardinality as C, then A has the same
 BS> cardinality as C (the transitive property of cardinality).
Basic.  Hence the cardinal numbers can be generated as equivalent classes of 
equinumerous sets.
 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.
I use the word denumerable to mean countably infinite.
 BS> One must understand the above to make sense of the rest of my proofs,
Learned this stuff while still in high school.
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