Question

defined countable and uncountable set and axioms of Choice​

Answer 1

Answer:

Step-by-step explanation:

In mathematics, an uncountable set (or uncountably infinite set)[1] is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions hold:

There is no injective function (hence no bijection) from X to the set of natural numbers.

X is nonempty and for every ω-sequence of elements of X, there exist at least one element of X not included in it. That is, X is nonempty and there is no surjective function from the natural numbers to X.

The cardinality of X is neither finite nor equal to {\displaystyle \aleph _{0}}\aleph _{0} (aleph-null, the cardinality of the natural numbers).

The set X has cardinality strictly greater than {\displaystyle \aleph _{0}}\aleph _{0}.

The first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.