 1. vsevolod_schenkevich says:

In set theory, we can compare infinite sets(i.e. infinities) as we compare finite sets with respect to the number of elements.

—————————–

Reference:

A set A is called a subset of B if any element of A is an element of B

Two sets are equally powerful if each element of the set A can be associated with a single element of the set B according to some law.�

card(X) is the cardinal number of the set X.

—————————–

We introduce such a characteristic for sets as a cardinal number. For finite sets, this will be the number of elements in the set.�

For infinite sets, the cardinal number is a generalization of the concept of the number of elements.

Although the cardinal numbers of infinite sets are not reflected in natural numbers, they can be compared.

If the set A is equal to the subset B, then card(A)

If the set B is equal to the subset A, then card(B)

If card(B)<=card (A) and card(A)

In this case, comparing cardinal numbers of infinite sets has the same properties as comparing real numbers.

—————————————-

In this way, we can compare infinite sets. So we can say that the infinity of real numbers is greater than, for example, natural numbers.� It is known that the cardinal number of natural numbers is the smallest possible among infinite sets.

P.S. Many people who first encountered set theory may be baffled by the fact that a set can be equally powerful to its part (a strict subset). For example, all rational numbers are equally powerful as natural numbers.