Actual Infinity in Reality?

[GrandizerII]

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

---

Let A be a set with a universal set U.   We denote the complement of a set as A^c where A^c=U–A , which is the set containing all elements x with the defining property that x is not in A.

---

Thanks. Your approach is definitely a better alternative. The important thing to take away from this is that Hilbert's Hotel is about set operations not merely number operations. Both infinity and "finity" (is this a word?) are not numbers, after all.