User blog comment:Forthright/QUICK QUESTION/@comment-39541634-20191026174000

ε₀ is not a number at all.

It is the first ordinal(*) which cannot be written by doing ordinary arithmetic (addition, multiplication and exponentiation) on ω(**).

(*) Ordinals are an extension of the ordinary numbers into the infinite realm. They represent increasingly complex infinite structures. Finite numbers (1,2,3,4,...) are also regarded as ordinals, even though they are finite.

(**) ω is the smallest infinite ordinal. It's what comes directly after all the finite numbers: 1,2,3,4,... . And after ω, we get (as you might expect) ω+1. This continues, welll... indefinitely. In fact, every time you say "and so on" you define a new ordinal.