Aleph-null or Aleph-one?

Number Theory Level 3

How many of these sets have a cardinality of $\aleph_1$ ?

A. Hilbert's Infinite Hotel
B. $\mathbb{Q}'$
C. $\mathbb{R}$
D. $\mathbb{Z}$

Assume the continuum hypothesis and the axiom of choice is true.

0 1 2 3 4

1 solution

Terence Chan
Jan 20, 2021

Since Hilbert's Hotel has countable rooms, its cardinality is aleph-null.

$\mathbb{Q}'$ is the set of irrational numbers, so it is uncountable and its cardinality is aleph-one.

$\mathbb{R}$ is the set of real numbers, so its cardinality is aleph-one because it includes the set of irrational numbers, which is uncountable.

$\mathbb{Z}$ is the set of integers and like Hilbert's Hotel, it is countable and so it's cardinality is aleph-null.

So, two of the given have cardinality aleph-one.