Bob Sewell on "Infinity"
with me...
 RT>> Wrong. There are half as many even numbers as there are integers.
 MB>>  Can you prove this?  
 RT>> Its a logical conclusion. The set of all even integers can never be as
 RT>> big as the set of all integers. It has to be half by definition.
 BS>     It seems logical because our minds are used to dealing with finite
 BS>  numbers, and half of any finite number is smaller than the finite
 BS>  number.
That is a a good point and one I haven't fully appreciated I must admit.
 BS> But in dealing with infinity, things change.  Please see my
 BS>  post from March 21st wherein I show that:
 BS>  infinity + infinity = infinity = infinity ^ 2 = infinity ^ infinity,
 BS> etc.
I don't dispute this Bob, but I was considering sets. Let me explain what I 
base my own statement on and why I see it as a logical conclusion.
We have a set of even integers, and we have a set of all integers. The first 
set is part of the second set and is delimited by that set. If the latter set 
does not exist then neither can the former set. DYSWIM?
You cannot make the set of all integers from the set of all even numbers. Why 
not?
Or am I misunderstanding set theory here?
 BS>     If it has scrolled off your system, tell me and I'll repost the
 BS>  proof for you.
No need, I marked it keep. :)
 MB>>  practical purposes.  Here is the proof. 1)  ((Infinity MOD 2) = 0)  =
 MB>> (0.5 * Infinity)   2)  (0.5 * Infinity) = Infinity,
 MB>> so also, 3)  ((Infinity MOD 1) = 0)  = Infinity  ,
 MB>> therefore:
 MB>> 4)  ((Infinity MOD 2) = 0) minus ((Infinity MOD 1) = 0) = 0
 MB>>  They are equivalent terms.
 RT>> Infinity MOD 2 = 0 is only true if infinity is an even number. Is it?
 RT>> If infinity is an odd number (which is equally possible) then Infinity
 RT>> MOD 2 = 0.5
 BS>     Well, an odd number MOD 2 = 1, not 0.5, but you are correct in that
 BS>  his proof isn't valid as stated because the oddness or evenness of
 BS>  infinity is indeterminate.
 I knew it was 1.
 BS> If you have a very durable lamp and at noon
 BS>  every day you turn its switch to toggle it off or on, will it be on or
 BS>  off after an infinity of days?  You can never know.
Precisely. That idea is what I based my answer on.
 BS>     Again, I refer you and Mark to my post on infinity from the 21st to
 BS>  show the proof of what Mark was saying, that the infinite set of even
 BS>  integers is the same size (cardinality) as the infinite set of all
 BS>  integers, because both sets are countably infinite and both have a
 BS>  one-to-one correspondence to the set of all positive integers.
This is where I'm confused then, because logically speaking the set of even 
numbers does not bear a one-to-one correspondence with the set of all 
integers.
 BS>     This is what I mean, and maybe this is all the proof you'll need.
 BS>  In any case, it shows you what countably infinite means and gives a
 BS>  visual image of my proof to go with the one I referred you to.
 BS>     Positive Integers:  1  2  3  4  5  6  7  8  9  10  11  12  13  ...
 BS>          All Integers:  0  1 -1  2 -2  3 -3  4 -4   5  -5   6  -6  ...
 BS>         Even Integers:  0  2  4  6  8 10 12 14 16  18  20  22  24  ...
Let's take the three lines as an example of three sets with the numbers you 
have actually shown as we'll have three sets containing the same amount of 
elements.
Where does the last element in the even-integer set, 24, appear in the 
all-integer set, or the positive-integer set?
Double the size of the sets and ask the same question.
And double them again... ad infinitum?
AIUI, infinity is not a number in itself rather an expression to indicate an 
endless sequence.
    Relatif Tuinn
... Beware of the Troll!
--- Spot 1.3a #1413
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