FOM: Re: Transfinite Logic

[wiman lucas raymond]

>It's clear you can correlate the even numbers with themselves
>perfectly, as follows:

>    1,2,3,4,5, ...
>    *,2,*,4,*, ...

>This way, there is a whacking great remainder, as indicated by the
>asterixes.  If equinumerosity is1-1 correlation without remainder, the
>even numbers are not equinumerous with the integers.  If on the other
>hand remainders are OK, what does the diagonal proof prove?

By this "logic", the natural numbers aren't equinumerous with
themselves.

Consider:

1, 2, 3, 4, 5, ...
*, 1, *, 2, *, ...

There is definitely a remainder, but it seems ridiculous to conclude
that the natural numbers don't have the same number of elements as the
natural numbers.  You seem to allow only "inclusion" functions for
testing for equinumerosity.  But if these are the only functions
allowed, then how do we prove that 
{1,2,3,4,5} and {6,7,8,9,10} have the same number of elements?

- Lucas Wiman