19 Oct '04 23:52

Hmm, it seems no-one got 2. Well, injections do exist from F(R) to R. For example:

- Put each number in the set through an injection R -> [0,1) by e.g. 'zigzagging' through the decimal expansion (so 123.456 becomes 0.342516);

- Interlace the resulting numbers by putting the first DP of the first, then the first of the second, etc (so 0.123, 0.45 and 0.6789 would combine to give 0.146257308009);

- Add n, the size of the set, so you get n point something or other.

Can anyone answer the general question, whether there exists an injection F(X) -> X for an arbitrary infinite set? Remember an arbitrary set X needn't have an ordering, addition or multiplication, and it could be a different size from both N and R.

- Put each number in the set through an injection R -> [0,1) by e.g. 'zigzagging' through the decimal expansion (so 123.456 becomes 0.342516);

- Interlace the resulting numbers by putting the first DP of the first, then the first of the second, etc (so 0.123, 0.45 and 0.6789 would combine to give 0.146257308009);

- Add n, the size of the set, so you get n point something or other.

Can anyone answer the general question, whether there exists an injection F(X) -> X for an arbitrary infinite set? Remember an arbitrary set X needn't have an ordering, addition or multiplication, and it could be a different size from both N and R.