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Does ZFC permit or acknowledge the existence of infinite sets that are uncountable?

By saying infinite sets that are uncountable, I mean that the cardinality of the sets is uncountable (that is $\ge\aleph_1$)

Thanks.

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    @Michael Hardy: The question was bigger than or equal to $\aleph_1$. This is the definition of uncountable in ZFC (where as in ZF it only means "not smaller or equal than $\aleph_0$").2012-02-11

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Cantor's theorem tells us that if $X$ is a countably infinite set then $P(X)$ is not countable.

In ZFC we assert that if $x$ is a set then $P(x)$ is a set as well, so if you assume the axiom of infinity which asserts that indeed an infinite set exists then there exists an uncountable set as well.

There is more to that as well: How do we know an $ \aleph_1 $ exists at all?

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    @user24796: To say that the real numbers cannot be characterized in FOL is to say that you cannot have a theory that every model which satisfies this theory is exactly the real numbers (namely complete and Archimedean field).2012-02-12
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$\mathbb{R}$ and $\mathbb{C}$ are constucted within ZF (so ZFC)

the answer is yes

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The set of real numbers $\mathbb{R}$ spring to mind. Another example is the power set of the natural numbers (which exists by the power set axiom of ZFC), and that is uncountable by Cantor's theorem.