Can we say that cardinality of $\lvert \mathbb{R}\rvert$ is equal to infinity?

Question

I am a bit confused with cardinality at the moment. I know that the cardinality of $\mathbb{R}$ is equal to $\lvert(0,1)\rvert$, but does that mean they are equal to infinity, if not what are they equal to ?

Answer 1

"Infinity" is not the name of a cardinality. It makes sense to say that a set is infinite, meaning that it's not finite. But infinite sets can have many different cardinalities. There's not really a special name for the cardinality of $\mathbb R$, other than "the cardinality of $\mathbb R$," or more traditionally "the cardinality of the continuum." It's sometimes denoted by $\mathfrak c$.

Answer 2

Their cardinality is referred to as "the cardinality of the continuum." There are infinitely many different infinite cardinal numbers. In many contexts, though, it enough to say that the cardinality is "uncountably infinite" or that the set is "uncountable." An example of a countable infinite set is the integers.

Answer 3

The cardinaility of $\mathbb{R}$ is infinite, but note that there are various forms of infinity.

We use the symbol $\aleph_0$ to denote $|\mathbb{N}|$, the cardinality of the set of natural numbers.

The symbol $2^{\aleph_0}$ denotes $|\mathbb{R}|$.

It is a famous theorem of Cantor that $2^{\aleph_0}> \aleph_0$.

(The smallest infinite larger than $\aleph_0$ is denoted $\aleph_1$. The continuum hypothesis is the claim that $2^{\aleph_0}=\aleph_1$. Neither it nor its negation can be proved using the usual axioms of set theory.)