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1. 1. Is there a useful number system with a greater cardinality than R?

As yeah in a sense, I don't see how something can be "bigger" than R(plus be semi useful mathematically). It would be nice to have an example that isn't P(R) i.e. the powerset of R or P(P(Q)) i.e. the power set of the powerset of rational numbers.

2. Is there something stronger than undecidability?

Is there something that can't be decided with an algorithm, but also can't be shown to be decidable by an algorithm? I think no, as if something can't be decided wouldn't that mean its true, as you can't find a counter example to it?
2. (Original post by Simplicity)
1. Is there a useful number system with a greater cardinality than R?
What are you happy considering as numbers? Do you consider for example functions as numbers?
3. (Original post by SimonM)
What are you happy considering as numbers? Do you consider for example functions as numbers?
Yeah, I just looked up beth numbers and it had some examples in wikipedia that isn't the powerset.

I suppose a function is sort of like a number.

Hmm, it says i.e. all function R to R.

But, I can't think of a example of a beth three. http://en.wikipedia.org/wiki/Beth_number

Is ?
4. Surreal numbers are so big they aren't even a set.

http://en.wikipedia.org/wiki/Surreal_number

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