User blog comment:Edwin Shade/Understanding The Infinite/@comment-30754445-20171109171549

ζ₀^ζ₀^ζ₀... is not ζ₁

Such power towers always give you the next epsilon number after the base. That's why ε₀^ε₀^ε₀^... is ε₁ and ε₁^ε₁^ε₁... is ε₂.

So, what's the next epsilon number after ζ₀? Well, recall that ζ₀ is defined as the smallest solution of the equation β = εᵦ. This means that:

ζ₀ = εζ₀ (yes, these two ordinals are indeed equal!)

and the next epsilon number after εζ₀ is:

ζ₀^ζ₀^ζ₀... = εζ₀+1

As for ζ₁, that's a much larger ordinal. It is the 2nd smallest solution to the equation equation β = εᵦ. Can you find a fundamental sequence for this ordinal (hint: it uses the same trick you've used in your fundamental sequence for ζ₀)?