Working with the set of real numbers

Marko Rauhamaa marko at pacujo.net
Thu Feb 13 23:00:37 CET 2014


Rotwang <sg552 at hotmail.co.uk>:

>>          for x in continuum(0, max(1, y)):
>>              # Note: x is not traversed in the < order but some other
>>              # well-ordering, which has been proved to exist.
>>              if x * x == y:
>>                  return x
>
> [...]
>
> More importantly, though, such a computer could not complete the above
> iteration in finite time unless time itself is not real-valued. That's
> because if k is an uncountable ordinal then there is no strictly
> order-preserving function from k to the unit interval [0, 1].

If you read the code comment above, the transfinite iterator yields the
whole continuum, not in the < order (which is impossible), but in some
other well-ordering (which is known to exist). Thus, we can exhaust the
continuum in ℵ₁ discrete steps.

(Yes, the continuum hypothesis was used to make the notation easier to
read.)


Marko



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