Ordinal Numbers

Abstract

The successor x ⁺ of a set x was defined as x ⋃ {x}, and then ω was constructed as the smallest set that contains 0 and that contains x ⁺ whenever it contains x. What happens if we start with ω, form its successor ω ⁺, then form the successor of that, and proceed so on ad infinitum? In other words: is there something out beyond ω, ω ⁺, (ω ⁺)⁺, ⋯, etc., in the same sense in which ω is beyond 0, 1, 2, ⋯, etc.?