RE: Actual Infinity in Reality?
February 15, 2018 at 2:04 am
(This post was last modified: February 15, 2018 at 2:09 am by Kernel Sohcahtoa.)
(February 15, 2018 at 1:19 am)Grandizer Wrote: About the seeming contradictions in Hilbert's Hotel, here's to put things in clearer perspective:
(1) inf(1,2,3,4,5,6,7,8,9,10,...) - inf(1,2,3,4,5,6,7,8,9,10...) = 0
In this case, inf(positive integers) - inf(positive integers) = 0
or
inf - inf = 0
(2) inf(1,2,3,4,5,6,7,8,9,10,...) - inf(2,4,6,8,10,...) = inf(1,3,5,7,9,...)
In this case, inf(positive integers) - inf(positive even integers) = inf(positive odd integers)
or
inf - inf = inf
(3) inf(1,2,3,4,5,6,7,8,9,10,...) - inf(4,5,6,7,8,9,10,...) = 3
In this case, inf(positive integers) - inf(positive integers except for 1, 2, and 3) = 3
or
inf - inf = 3
So no contradictions, just different infs we're dealing with.
Also, case 1 proves there is no contradiction (because same collection - same collection is indeed 0).
Either way, without context, that is why inf - inf is indeterminate, much like 0/0 is indeterminate.
Regarding the examples posted above, another way to see it is to define the positive integers as the universal set. Thus, when we take the complement of each of the examples (the set of positive integers, the set of positive even integers, and the set containing 4,5,6,7,8,9,10,...) we obtain the empty set, the set containing the positive odd integers, and the set containing 1,2,and 3 respectively.
P.S. For anyone interested