How is aleph-one related to aleph-null?

In mathematics, aleph-null (${\aleph }_0$) and aleph-one (${\aleph }_1$) are transfinite cardinal numbers used to measure the size of infinite sets. Their relationship is defined by their order and magnitude:12

• Aleph-null (${\aleph }_0$): This is the smallest transfinite cardinal number, representing the cardinality of the set of natural numbers (or any set that can be put into a one-to-one correspondence with them). Sets with this cardinality are described as countably infinite or denumerable.314
• Aleph-one (${\aleph }_1$): This is defined as the next cardinal number after aleph-null, meaning it is strictly "larger" than ${\aleph }_0$.15

The relationship between these two numbers is central to the continuum hypothesis, which conjectures that the cardinality of the real numbers—often denoted as c—is equal to ${\aleph }_1$. This hypothesis asserts that there are no sets with a cardinality strictly between that of the natural numbers and the real numbers. While this remains a foundational question in set theory, it has been proven to be undecidable within the standard ZFC axiom system.125