Can there be numbers with $ℵ_n$ digits?(n>0)

Numbers can have their cardinality of the number of digits only over the set of natural numbers i.e. $\mathbb{N} $ that is because each digit has an ordinal position. So the greatest cardinality of the number of digits is $\aleph_0$.

On the other hand, there is a surreal number system which explores higher cardinalities.

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Number Theory

Cardinals

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