On Mon, Sep 16, 2024 at 9:36 PM Alan Grayson <[email protected]> wrote:
> How would you map (0,1) 1-1 onto the real numbers? > *F(x)=1/2 + 1/π Arctan(x) . The domain is all the real numbers and the range is (0.1)* *> **Both sets have the same cardinality.* > * And a short line in a long line have the same cardinality, and there is a very simple geometric proof showing that they can be put into a one to one correspondence. Just draw a triangle with a line from the apex running through the short line to the long line at the base of the triangle * *> If a set is infinitely large then there is a proper subset of that set > that can be put into a one to one correspondence with the entire set, in > fact that is the mathematical definition of "infinity".* > *> That's circular. AG * *Alan, if you open any book on set theory you will find that the formal mathematical definition of infinity is that something is infinite if and only if a proper subset of it can be put into a one to one correspondence with the entire thing. * John K Clark See what's on my new list at Extropolis <https://groups.google.com/g/extropolis> ioi -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CAJPayv2QYCGruQvWUcUNnc99nv24iEAhCtbb1AXdNoC77x3S-w%40mail.gmail.com.
