You can only compare finite collections in that way. A simple example, take all the numbers and then take, all the even numbers, and align them like this:
1 <> 2
2 <> 4
3 <> 6
4 <> 8
...
Then there are as many even numbers as there are all the numbers.
You can easily make that into a sequence of points not going to infinity like this:
1 <> 1/2
1/2 <> 1/4
1/3 <> 1/6
1/4 <> 1/8
...
Then the second sequence is a subset of the first, but they have the "same number of points"
So - it's like that. This method of lining up only works when the two collections concerned are finite.
For another simpler example - whenever you make a map - then you are making a similar one to one correspondence of a larger area onto a smaller area and you can draw similar lines lines from the one to another.
Or just draw a line a of length 1 next to a line b of length 2.
And for any point x on line a, join it to the point 2x on line b.
1 <> 2
2 <> 4
3 <> 6
4 <> 8
...
Then there are as many even numbers as there are all the numbers.
You can easily make that into a sequence of points not going to infinity like this:
1 <> 1/2
1/2 <> 1/4
1/3 <> 1/6
1/4 <> 1/8
...
Then the second sequence is a subset of the first, but they have the "same number of points"
So - it's like that. This method of lining up only works when the two collections concerned are finite.
For another simpler example - whenever you make a map - then you are making a similar one to one correspondence of a larger area onto a smaller area and you can draw similar lines lines from the one to another.
Or just draw a line a of length 1 next to a line b of length 2.
And for any point x on line a, join it to the point 2x on line b.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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