Axiom of Infinity

Set Theory · Axiom Academy

Before stating the axiom, we need to define the successor of a set. For any set x, we define its successor as S(x) = x ∪ x . The successor "adds one more element" to a set - it takes the set x and adjoins x itself as a new element. This operation is key to building the natural numbers. The axiom states: There exists a set I that contains the empty set and is closed under the successor operation. In other words, if x is in I, then S(x) = x ∪ x is also in I. This gives us: ∅ ∈ I, S(∅) ∈ I, S(S(∅)) ∈ I, and so on forever. 3. Constructing ω (the Natural Numbers) We identify natural numbers with sets: 0 = ∅, 1 = ∅ , 2 = ∅, ∅ , 3 = ∅, ∅ , ∅, ∅ , and so on. The set ω is defined as the smallest infinite set satisfying the Axiom of Infinity - the intersection of all such sets. This is precisely the set of natural numbers!

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