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Countable sets have the same cardinality as the natural numbers (or a subset of the natural numbers, depending on who you ask).

Can one make the same claim about uncountable sets and real numbers?

In other words, do all uncountable sets have same cardinality as real numbers?

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For every set $A$ we have that $\mathcal P(A)$ has a strictly larger cardinality. Therefore $\mathcal P(\Bbb R)$ has a strictly larger cardinality than that of $\Bbb R$, and is therefore uncountable as well.

No. One of the fundamental results of set theory is Cantor’s theorem, which states that for any set $X$, the set of all subsets of $X$ (AKA the power set of $X$) always has a greater cardinality than $X$ does. So in particular, the set of all subsets of the set of real numbers has a greater cardinality than the set of real numbers.

By the way, Cantor’s theorem is actually one way to show that the set of real numbers has greater cardinality than the set of natural numbers, because it can be easily shown that the set of real numbers has the same cardinality as the set of all subsets of the set of natural numbers. (Try proving this.)

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