On 3/18/2013 7:29 AM, WM wrote: > > Every list yields another diagonal. > And every diagonal can be included in another list. > > Both these facts are the two sides of infinity.
What these facts correspond with is the structure of a directed set.
Directed sets are, indeed, infinite.
Absolute infinity cannot be treated as a number.
That does not mean one cannot have a transfinite arithmetic.
> With same right he could have argued that there are more even than odd > non-negative integers, because 0 is even and after every odd integer > there follwos an even one, just overlooking that after every even > integer there follows an odd one.
But, Cantor understood the strict reasoning inherent to the nature of class-based logical construction.
What he could have done if he were of lesser skill is not what he did do -- because he knew better.