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Math & Stuff:

Some infinities are bigger than other infinities. A writer we used to
like taught us that. There are days, many of them, when I resent the
size of my unbounded set. I want more numbers than I'm likely to get,
and.... I cannot tell you how thankful I am for our little infinity. I
wouldn't trade it for the world. You gave me a forever within the
numbered days, and I'm grateful.”

―John Green, *The Fault in Our Stars*

The proof:

Cute story, but the video is wrong. There is no end to the counting numbers, therefore one cannot run out of them.

ReplyDeleteHow about this proof? Consider the function y=x, it's a straight line passing through the origin at a 45 degree angle. Let's use calculus to find the area under the line from 0 to 1. We will approximate the area by summing an infinite number of small rectangles, each rectangle having an area = xdx and sum these up from 0 to 1. The integral of x is (x^2)/2 and the area is 1/2 under the line from 0 to 1. Now we do the same thing from 0 to 2 instead and find that the area is 2. Since 2>1, the infinity of rectangles from 0 to 2 must greater than the infinity between 0 and 1.

DeleteBy its definition infinity cannot be measured. How then can one infinity be greater than another?

ReplyDeleteIt's not measuring, it's one being larger than another. What if you you have y=x divided by y=x^2, and determine the value as x approaches infinity? It goes to zero, because the denominator becomes larger so much faster. Isn't one one infinity greater than the other?

ReplyDeleteSee Some Infinities Are Larger Than Others: The Tragic Story of a Math Heretic at http://www.wired.com/2007/07/some-infinities/

Nice.

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