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On 13/02/2014 22:00, Marko Rauhamaa wrote: > 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 >> >> [...] Restoring for context: >>> The function could well return in finite time with a precise result >>> for any given nonnegative real argument. >> 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 ?? discrete steps. Yes, I understood that. But my point was that it can't carry out those ?? discrete steps in finite time (assuming that time is real-valued), because there's no way to embed them in any time interval without changing their order. Note that this is different to the case of iterating over a countable set, since the unit interval does have countable well-ordered subsets.

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