Christoph writes:
> In addition, wavefunctions can be seen as functions over 
> space and time, so that the minimum measureable intervals 
> which make it impossible to say that space and time are sets,
> allow to deduce that it is impossible that wavefunctions
> form sets. (At Planck scales, of course)
>
> In particular, at Planck scales, wavefunctions do not form
> Hilbert spaces. (In fact, it is unclear whether wavefunctions
> make sense at all in these conditions.)

Right, the real point is that our current theoretical models do not
extend to the Planck length.  We don't have a theory of quantum gravity.
Under those circumstances, we can only speculate whether an eventual
theory will be based on some type of continuum, on discrete logic,
or something else.

The common popular descriptions I have seen of Planck scale spacetime
is of a churning foam, with Planck length wormholes constantly popping
into and out of existence, with the local topology very poorly defined.
Maybe some kind of fractal dimensionality.  It's not a simple discrete
lattice, with integers representing space and time coordinates.

I don't know what exactly it would mean to say that this is not a "set"
(a set is an extremely broad concept), but in any case without a theory
we really can't say anything at all about it.

Hal

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