🚀 go-pugleaf

Tosser wrote:
> 
> Charles R. Rutledge wrote:
> 
> > Sorry, Guy.  Both the set of all integers and the set of even integers
> > are the some size.  For every integer, you can name a corresponding
> > even integer (2 X the integer).  Since this is a one-to-one and onto
> > relationship,  the two infinities are equal.  Isn't god wonderful???
> >
> > Charlie
> 
> In another response to this thread, I wrote:
> 
> "It has something to do with both of them being "countably infinite",
> since you can make direct correspondences between members of each set,
> i.e.  1 with 1 AND -1, 2 with 2 AND -2,...ad infinitum."
> 
> OKAY, MATH GEEKS:  Who's right, (if anyone) and why?
> 
> T

Both of you are right.  The definition of "countably infinite" is that 
the set can be mapped onto the set of natural number, which is the same 
as the nonnegative integers.  There is no larger set of integers, because 
you can create a rule that maps any set of integers onto the set of 
natural numbers, or a subset thereof.

By contrast, between 1 and 2, there are an infinite number of rational 
numbers -- just in the series 1/2, 2/3, 3/4, 4/5, 5/6, ... there is a 
countably infinite number of members, and there are an infinite number of 
such series of rational numbers between every two integeres.  Hence, the 
rational numbers cannot be mapped onto the natural numbers, and is called 
"uncountably infinite."

Worse, between each pair of rational numbers are an infinite number of 
irrationals, numbers like pi/4, e/3, 5^(1/2), and so forth; this makes 
the real numbers (the set that includes all integers, all rational 
numbers, and all irrational numbers) a still larger infinity than the 
uncountable set of rational numbers.

I gather that mathematicians have enumerated a (to my mind) rather 
embarassingly long list of such further infinities.  Infinity, as far as 
I'm concerned, just isn't what it used to be...  B)

-- 
A confidence man knows he's lying; that limits his scope.  But a
successful shaman believes what he says -- and belief is contagious;
there is no limit to >his< scope.  -- Jubal Harshaw, M.D, J.D.

Donald Qualls, aka The Silent Observer           NAR # 70141-SR Insured
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Opinions expressed are my own -- take them for what they're worth
and don't expect them to be perfect.