Why Is Australia Not Considered An Island ??

Why Is Australia Not Considered An Island ?? Jan 9, 2020 22:37:53 GMT

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Post by general313 on Jan 9, 2020 22:37:53 GMT

Dec 23, 2019 4:14:11 GMT Jep Gambardella said:

Dec 23, 2019 2:30:37 GMT Hairynosedwombat said:

George Cantor might disagree with you here. You imply that there are more rectangles than squares.

We can map (or make a one-to-one correspondence) between each member of the set of all rectangles and the set of all natural numbers (ie 1, 2, 3...).

We can make the same mapping of all squares with the natural numbers.

Therefore I would suggest (and I believe Cantor proved) that the set of all rectangles and the set of all squares are the same size.

That is ridiculous. Obviously there are more rectangles than squares. Both numbers are infinite, but one is bigger than the other - just like the number of multiples of 3 is bigger than the numbers of multiples of 5 (even though both are infinite).

Actually multiples of 3 can be put into one-to-one correspondence with multiples of 5, and therefore have the same count. This is easy to verify. What's a bit more mind-blowing is that Cantor showed that the points on a line are the same in number as the points in a plane, and therefore their count is the same.