22 Jun 2016, 1:09 pm
Tollorin
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Joined: 14 Jun 2009
Age: 42
Gender: Male
Posts: 3,178
Location: Sherbrooke, Québec, Canada
23 Jun 2016, 2:13 pm
eric76 wrote:
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|.
Even then, there is still infinite that are bigger that other. Take the natural numbers; 1,2,3,4,5,6... and so on. There is a infinity of natural numbers. Yet, only by adding a decimal; 0.1, 0.2, 0.3, 0.4, 0.5, .... and so on, you are multiplying that infinite by ten. And you can multiply this result by ten a infinite number of times. There is a infinity of infinite.
23 Jun 2016, 5:14 pm
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