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Originally Posted by tw
We have two sets. Set A = {1, 2, 4, 8, 16, 32 ...}. Set B = {2, 4, 8, 16, 32, 64 ...}. Two examples of infinity. But to be equal, then 1 = 2; 2 = 4; 4 = 8; etc. Clearly they are not equal. IOW we have two different sizes of infinity.
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No, to be equal
in size, there must be a one-to-one
correlation between terms, 1 --> 2, 2 --> 4, 4 --> 8, etc. The infinte sets you listed are equal in size. Doesn't matter that their terms are different, for every next term in set A there will be exactly on next term in set B.
An example of a larger infinity would be a "set of sets," i.e, {Set A, Set B, Set C...} where each set is also infinite. However, and here's where it gets wacky, this set
can be the same size as a normal infinte set
if you count it the right way.
Think of it as a grid like the first picture below. If you start counting down one column, you'll be going downward for infinity and never get to column B. BUT, if you count back and forth along the diagonals as shown in the second picture, you can reach infinity in both the horizontal and vertical directions. Thus the set of all infinite sets does have a one-to-one correlation with an infinite set as long as you count it this particular way.
This is the same as the part of the Hotel Infinity story where an infinite nunbedr of buses arrive each with an infinite number of people, assuming the website is using the same allegory my professor used.