I Countable or Uncountable Cardinality of Multiverse?

[FallenApple]

In the MWI, are the number of universes in the multiverse countable or uncountable? It seems like if all possibilities happen, then that is like the power set, which has uncountable cardinality. Or maybe a Cantor diagonalization argument can be used on the discrete sequence of events over the discrete time intervals(maybe Plank time?).

 

[andrewkirk]

I think it would depend, amongst other things, on whether the universe is spatially and temporally finite. If it is both, and we can use some sort of Planck-timey / Planck-lengthy thing to discretise spacetime into a very large finite set of cells, then the powerset of that will be even larger, but still finite.

If it is either spatially or temporally infinite and in an infinite sub-collection of cells there is more than one possibility, then a full multiverse of alternatives be uncountable.

If we can't discretise spacetime, so that there are an infinite number of times within a one-second interval at which a nucleus could emit a neutron then it will bbe uncountable.

 

[FallenApple]

I think it would depend, amongst other things, on whether the universe is spatially and temporally finite. If it is both, and we can use some sort of Planck-timey / Planck-lengthy thing to discretise spacetime into a very large finite set of cells, then the powerset of that will be even larger, but still finite.

If it is either spatially or temporally infinite and in an infinite sub-collection of cells there is more than one possibility, then a full multiverse of alternatives be uncountable.

If we can't discretise spacetime, so that there are an infinite number of times within a one-second interval at which a nucleus could emit a neutron then it will bbe uncountable.

So it seems like it requires a very specific set of circumstances just for countability to be true. If there is Uncountability, and MWI is true, then we have a bijection with the real numbers. Then the concept of numbers such as $\pi$ could be physically realized in its full entirety. Or even non computable numbers.

 

[andrewkirk]

If there is Uncountability, and MWI is true, then we have a bijection with the real numbers.

Not just a bijection, but a surjection from the universes to the reals. Under some models they could have a greater cardinality than the reals.

Then the concept of numbers such π\pi could be physically realized in its full entirety.

I'm trying to think what such a realisation might be. We can't have a 'perfect circle' because that would require lines of zero width. But perhaps an electromagnetic wave, one cycle of which takes exactly $\pi\times 10^{-9}$ seconds? Sounds conceivable in an uncountable model.

 

[FallenApple]

Not just a bijection, but a surjection from the universes to the reals. Under some models they could have a greater cardinality than the reals.
I'm trying to think what such a realisation might be. We can't have a 'perfect circle' because that would require lines of zero width. But perhaps an electromagnetic wave, one cycle of which takes exactly $\pi\times 10^{-9}$ seconds? Sounds conceivable in an uncountable model.

Ah, so if it's a surjection, then there are some universes that can't even be uniquely labelled with a real. So an uncountable multiverse would be truly too large to be described by numbers. Set theory would be needed.

The cycle makes sense. It can be mapped to a circle under continuous time, even if space is discrete, using the set of all same photon over parallel universes over the time interval.

Also, maybe another physical realization is the set of the same n-gon but in uncountability parallel universes under different continuous rotation. The set as a whole would exhibit $\pi$. Maybe such an n-gon would have to be a molecule or something. Or if the continuous rotation isn't possible, maybe expand the set to different n-gons, same object but in nonparallel branches, to fill in the gaps. And a sphere could be the uncountably infinite rotational positioning of say an icosahedron structure.