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 said: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.
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.FallenApple said:If there is Uncountability, and MWI is true, then we have a bijection with the real numbers.
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 said:Then the concept of numbers such π\pi could be physically realized in its full entirety.
andrewkirk said: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.
The concept of countable and uncountable cardinality refers to the size or quantity of objects or elements within the multiverse. Countable cardinality means that the number of objects or elements can be counted and represented by a natural number (1, 2, 3, etc.). Uncountable cardinality means that the number of objects or elements is infinite and cannot be represented by a natural number.
The countable or uncountable cardinality of the multiverse is determined by the number of possible universes or realities that exist within it. If the number of universes is finite and can be counted, then the multiverse has a countable cardinality. If the number of universes is infinite and cannot be counted, then the multiverse has an uncountable cardinality.
Yes, it is possible for both countable and uncountable cardinality to exist within the same multiverse. This would mean that there are both a finite number of universes and an infinite number of universes within the multiverse.
The implications of countable or uncountable cardinality for the multiverse theory depend on the specific interpretation of the theory. For example, if the multiverse is considered to be a single entity with a finite number of universes, then a countable cardinality would support this interpretation. On the other hand, an uncountable cardinality could suggest the existence of an infinite number of separate and distinct multiverses.
The concept of countable and uncountable cardinality highlights the complexity and diversity of the multiverse. It suggests that there may be a vast range of possibilities and variations within the multiverse, and that our understanding of it may be limited by our ability to comprehend and quantify such vastness.