naturalplastic wrote:
zer0netgain wrote:
Or the mindbender that a subset of infinity is itself infinite even though it's smaller.
The set of odd numbers is smaller than the set of whole numbers. But both are infinity.
It is very easy to map the set of whole numbers onto the odd numbers and the odd numbers onto the whole numbers.
Therefore, there is a 1:1 correspondence between the two sets and thus both have the same number of elements.
Consider this:
The odd numbers are the positive integers not divisible by 2: { 1, 3, 5, 7, 9, ... }
The Whole numbers are the postive integers and 0: { 0, 1, 2, 3, 4, 5, ... }
Let O be the set of odd numbers and W be the set of whole numbers.
Define f :W->O by f(x)=2x+1 for all x in W. Since for every x in W, there is an f(x) in W, we have that |O| >= |W|.
Define g:O->W by g(y)=(y-1)/2 for all y in O. Since for every y in O, there is a g(y) in W, we have that |O|<=|W|.
Thus, |O|=|W|.