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