Axiomatic Set Theory: Unveiling the Foundations of Mathematics

The Axiom of Separation states that for any set (A) and any property (P(x)), there exists a set (B) whose elements are exactly the elements (x) of (A) that satisfy the property (P(x)).

The Axiom of Empty Set states that there exists a unique set with no elements, called the empty set and denoted by (\emptyset).

The Axiom of Union states that for any two sets (A) and (B), there exists a set (C) whose elements are exactly the elements of (A) and (B).

The Axiom of Power Set states that for any set (A), there exists a set (P(A)) whose elements are all the subsets of (A).

The Axiom of Infinity states that there exists a set (\mathbb{N}) whose elements are the natural numbers (0, 1, 2, \ldots).

The Axiom of Replacement states that for any set (A) and any function (f) from (A) to another set (B), there exists a set (C) whose elements are exactly the images (f(x)) of the elements (x) of (A).

The Axiom of Power Set states that for any set (A), there exists a set (\mathcal{P}(A)) whose elements are all the subsets of (A) and (\mathcal{P}(A)) is a set.

The Axiom of Universal Set states that there exists a set (V) that contains all sets.

The Axiom of Power Set states that for any set (A), there exists a set (\mathcal{P}(A)) whose elements are all the subsets of (A).

The Axiom of Choice states that for any collection (\mathcal{A}) of non-empty sets, there exists a function (f) from (\mathcal{A}) to the union of all the sets in (\mathcal{A}) such that (f(A)) is an element of (A) for each (A) in (\mathcal{A}).

The Axiom of Power Set states that for any set (A), there exists a set (\mathcal{P}(A)) whose elements are all the subsets of (A).

The Axiom of Universal Set states that there exists a set (V) that contains all sets.

The Axiom of Power Set states that for any set (A), there exists a set (\mathcal{P}(A)) whose elements are all the subsets of (A).

The Axiom of Choice states that for any collection (\mathcal{A}) of non-empty sets, there exists a function (f) from (\mathcal{A}) to the union of all the sets in (\mathcal{A}) such that (f(A)) is an element of (A) for each (A) in (\mathcal{A}).

The Axiom of Power Set states that for any set (A), there exists a set (\mathcal{P}(A)) whose elements are all the subsets of (A).