# Are ordinals greater than $\varepsilon_0$ used outside Ordinal Analysis?

I know of Conway's use of ordinals to exhibit the algebraic closure of $$\mathcal{F}_2$$. I also read a document about the Cantor Bendixson rank of some family of groups. But I found no applications of ordinals outside Ordinal Analysis that use "big" ordinals. In particular, no applications that make explicit use of notation systems like Veblen or Bachmann (or stronger). Are there any?

Edit: I already got useful answers (in the comments) in computability theory and in topology. However, I'll leave the answer open because I'm interesting in getting as many nice applications as possible.

I'm particularly interested in applications in "less abstract" branches of mathematics and, I explained earlier, in an explicit use of ordinal notations.

• In general topology, uncountable ordinals are often used in constructing counterexamples. Not sure if that's included in your notion of "big" ordinals. – Daniel Schepler Oct 12 '18 at 23:42
• you are correct but it doesn't seem satisfying as a complex notation system is not being used. (Are you referring to $\omega_1$ and other regular ordinals? Or are there stranger examples?) – Guillermo Mosse Oct 12 '18 at 23:48
• Yes, I was mostly referring to $\omega_1$ and ordinals "close" to that - and I think maybe in one example, I remember $\omega_2$ showing up. – Daniel Schepler Oct 12 '18 at 23:54
• Ordinals vastly bigger than $\epsilon_0$, but still countable, show up in computability theory and set theory quite frequently. E.g. $\omega_1^{CK}$, the least noncomputable ordinal, plays a very important role in computability theory. – Noah Schweber Oct 13 '18 at 0:06
• @NoahSchweber, cool answer. If you want you can answer it and I accept it. Nevertheless, I think I wasn't clear enough in my question (and it's my fault). I'll edit it. – Guillermo Mosse Oct 13 '18 at 1:36

How about googology? Pretty much all of this field is dedicated to creating bigger numbers than anyone else. And there are methods that people have used ordinals for to create numbers. I’m going to quickly explain the fast growing hierarchy as an example.

Every recursive limit ordinal has a canonical sequence, for example: the ordinal $$\omega$$ is the supremum of 1, 2, 3 and so on, $$\omega2$$ is the supremum of $$\omega$$, $$\omega+1$$, $$\omega+2$$ (where of course + n is the nth iteration of the successor function), even $$\varepsilon_0$$ has the canonical sequence $$\omega, \omega^\omega, \omega^{\omega^\omega}$$ and so on.

If we define $$f(a)$$ (where $$a$$ is a natural number) as $$a + 1$$, and then use recursion to define $$f^2(a)$$ as $$f(f(a))$$ with iteration continuing like expected. We can’t now define $$f_1(a)$$ as $$f^a(a)$$, and further: $$f_{b+1}(a) = f^a_b(a)$$. Now if instead of $$b$$ being a natural number, we use a limit ordinal, such as $$\omega$$, we replace the limit ordinal with the $$a$$th term in its canonical sequence. So for $$f_\omega(5)$$ we could reduce this to $$f_5(5)$$, and for $$f_{\varepsilon_0}(3)$$ we get $$f_{\omega^{\omega^\omega}}(3)$$

We can see how useful this would be for creating large numbers, as using large ordinals, like the Bachman-Howard ordinal would lead to extremely large numbers being created. Also, just in case you didn’t get my explanation of it, go to https://en.m.wikipedia.org/wiki/Fast-growing_hierarchy for a better explanation.

This is one (absolutely pointless, yet fun) use for large countable ordinals.

Although, this doesn’t just work for countable ordinals, it would work for ordinals such as $$\omega_1$$ too, if our mortal minds could comprehend its canonical sequence...

• Do you have a reference for fast-growing hierarchies with uncountable ordinals? – Keshav Srinivasan Nov 13 '19 at 6:05
• @KeshavSrinivasan The trouble with ordinals such as $\omega_1$ is that we can’t give them fundamental sequences. People have tried with non-recursive ordinals (which I forgot to exclude in my answer...) googology.wikia.org/wiki/User_blog:Deedlit11/… but uncountables are more difficult. Our best bet would be to collapse uncountables into extremely large countable (and recursive) ordinals, but like I said, if we can find a fundamental sequence for and ordinal (and for each of its fundamental sequence and so on) we can... – L. McDonald Nov 14 '19 at 2:43
• define numbers using it in the fast growing hierarchy. – L. McDonald Nov 14 '19 at 2:44