Set theory: fixed points of $n \mapsto \varepsilon_n$ and $n \mapsto \omega_n$

Question

For an ordinal number $\alpha$, the epsilon number $\varepsilon_\alpha$ is defined as the "$\alpha$-th" fixed point of the map $n \mapsto \omega^n$, i.e. $\omega^{\varepsilon_\alpha} = \varepsilon_\alpha$.

Question: What about fixed points of $n \mapsto \varepsilon_n$ or $n \mapsto \omega_n$? Examples for the former would be $\varepsilon_{\varepsilon_{\varepsilon_{\dots}}}$ and (if I'm not mistaken) all $\omega_\alpha$ for $\alpha\ge 1$, and an example for the latter would be $\omega_{\omega_{\omega_{\dots}}}$. Do they have names? Are they studied? Do they have interesting properties?

Answer 1

The least fixed point of $\mu \mapsto \varepsilon_\mu$ is called $\zeta_0$ and is written as $\varphi_2(0)$ using the Veblen $\varphi$ function. It is also equal to $\psi(\Omega)$ in Madore's psi function and $\psi_0(\psi_1(\psi_1(0)))$ in Buchholz's psi function. You can read more about it here: https://googology.fandom.com/wiki/Cantor's_ordinal

If by $\omega_\mu$ you mean the enumeration of infinite cardinals, then the least fixed point of $\mu \mapsto \omega_\mu$ doesn't have a specific name, but it is written as $\Phi_1(0)$ using Rathjen's $\Phi$ function. The ordinal doesn't have that many specific properties, apart from $\psi_{\chi_0(0)}(\Phi_1(0))$ in Rathjen's psi function being the countable limit of the extended Buchholz's psi function.

Answer 2

As Noah Schweber mentioned, the Veblen function is relevant for fixed points of $\alpha\mapsto\varepsilon_\alpha$. The Veblen hierarchy starts with $\varphi_0(\alpha)=\omega^\alpha$, and each $\varphi_\beta(\alpha)$ with $\beta>0$ enumerates the simultaneous fixed points of all the functions $\varphi_\gamma$ with $\gamma<\beta$.

$\varphi_n$ for finite $n$ can be proven total by the fixed point lemma, which says for any increasing function $f:\textrm{Ord}\rightarrow\textrm{Ord}$ which commutes with suprema (i.e. $f(\textrm{sup}(A))=\textrm{sup}(f''A)$ for $A\subset\textrm{Ord}$.), there exist arbitrarily large fixed points of $f$. In particular, we can change $\varphi_0$ to any normal function and each $\varphi_n$ remains normal, so the fixed-point lemma still applies. $\alpha\mapsto\omega_\alpha$ is a normal function, if we change $\varphi_0$ to this we get Rathjen's $\Phi_n$ functions with $n<\omega$. (To prove each function in the full $\Phi$ hierarchy is total, we need more than the fixed point lemma.)

Answer 3

There is a natural way of going beyond the operations addition, multiplication, exponentiation of ordinals to higher level operations, the next being tetration (but not the usual notion of tetration; see below) and hyper-tetration, the latter being a term I made up around 1990 but probably not used online until this 4 March 2002 sci.math post. In an analogous way that exponentiation is not suitable for giving explicit formulas to the terms of the transfinite sequence of $\varepsilon$-numbers, ordinal tetration is also not suitable. However, explicit formulas for these terms can be given using ordinal hyper-tetration.

In my answer to the MSE question Why are tetrations not useful? I discussed ordinal tetration a bit. Because that MSE question was closed and thus might not be visible to everyone, I'll repeat the definition and a few relevant comments given there.

Ordinal Tetration. Fix an ordinal $\alpha$. We define $\, \sideset{_{}^\beta}{}\alpha \,$ by transfinite induction on $\beta$ as follows.

(base case) $\;\; \sideset{_{}^0}{}\alpha = 1\;$ and $\; \sideset{_{}^1}{}\alpha = \alpha$

(successor case) $\;\; \sideset{_{}^{\beta + 1}}{}\alpha \; = \; \left(\sideset{_{}^{\beta}}{}\alpha \right)^{\alpha} \;$ for each $\; \beta \geq 1$

(limit case) $\;\; \sideset{_{}^{\lambda}}{}\alpha \; = \; \sup \left\{\sideset{_{}^{\beta}}{}{\alpha}: \; \beta < \lambda \right\}\;$ if $\; \lambda \;$ is a nonzero limit ordinal

Ordinal Tetration vs Usual Tetration. In the case of finite ordinals (i.e. non-negative integers), this is NOT the same as the usual tetration operation. For instance, $$\sideset{_{}^4}{}\alpha \; = \; \left( \sideset{_{}^3}{}{\alpha} \right)^{\alpha} \; = \; \left( \left( \sideset{_{}^2}{}{\alpha} \right)^{\alpha} \right)^{\alpha} \; = \; \left( \left( {\alpha}^{\alpha} \right)^{\alpha} \right)^{\alpha} \; = \; {\alpha}^{{\alpha}^{3}} $$ In fact, it can be shown that ("top-down" evaluations are used here) $$\sideset{_{}^{\epsilon_0}}{}{\epsilon_0} \; = \; {\epsilon_0}^{{\epsilon_0}^{\epsilon_0}} $$ Moreover, we also have $\; \epsilon_0 \; = \; \sup\left\{\omega, \; \sideset{_{}^{\omega}}{}{\omega}, \; \sideset{_{}^{\sideset{_{}^{\omega}}{}{\omega}}}{}{\omega}, \; \ldots \right\}$, and hence it follows that $$ \epsilon_0 \; = \; {\omega}^{{\omega}^{{\omega}^{{\cdot}^{{\cdot}^{\cdot}}}}} \; = \; \sideset{_{}^{\sideset{_{}^{\sideset{_{}^{\sideset{_{}^{\sideset{_{}^{\cdot}}{}{\cdot}}}{}{\cdot}}}{}{\omega}}}{}{\omega}}}{}{\omega} $$ (In the above, each ellipsis signifies the supremum of all finite-iterated exponentiation/tetration expressions evaluated "top-down".)

Strong Tetration and Weak Tetration. There does not seem to be a standard term for this distinction in the literature, probably because this distinction does not arise very often. Mark Neyrinck’s May 1995 undergraduate thesis An Investigation of Arithmetic Operations uses the term top-down and bottom-up. I propose, when these two types of tetration are being discussed together for natural numbers, to use the term strong tetration for the ordinary notion of tetration and the term weak tetration for the ordinal notion of tetration as defined above.

We now consider the next higher level operation (not mentioned in that MSE answer), which I'll call "hyper-tetration". This is also defined in a "bottom-up" manner.

Ordinal Hyper-Tetration. Fix an ordinal $\alpha$. We define $\, \sideset{_{}^{\{\beta\}}}{}\alpha \,$ by transfinite induction on $\beta$ as follows.

(base case) $\;\; \sideset{_{}^{\{0\}}}{}\alpha = 1\;$ and $\;\sideset{_{}^{\{1\}}}{}\alpha = \alpha$

(successor case) $\;\; \sideset{_{}^{\{{\beta + 1\}}}}{}\alpha \; = \; \sideset{_{}^{\alpha}}{}{\left(\sideset{_{}^{\{\beta\}}}{}\alpha \right)} \;$ for each $\; \beta \geq 1$

(limit case) $\;\; \sideset{_{}^{\{\lambda \}}}{}\alpha \; = \; \sup \left\{\sideset{_{}^{\{\beta \}}}{}{\alpha}: \; \beta < \lambda \right\}\;$ if $\lambda$ is a nonzero limit ordinal

The following results are proved in some personal notes of mine that I first wrote around 1990, and which I have revised several times since then:

Let $\;\alpha \geq 2.\;$ The least $\varepsilon$-number greater than $\; \alpha \;$ is $\; \sideset{_{}^{\{\omega\}}}{}\alpha.$

Explicit Expressions for $\varepsilon$-Numbers Using Hyper-Tetration:

1. $\;$Let $\;0 \leq n < \omega.\;$ Then $\;\varepsilon_n \; = \;\sideset{_{}^{\{\omega \cdot (n+1)\}}}{}{\omega}.$
2. $\;$Let $\;\beta \geq \omega.\;$ Then $\;{\varepsilon}_{\beta} \; = \; \sideset{_{}^{\{\omega \cdot \beta\}}}{}{\omega}.$

These two formulas can be replaced with the single formula $\;{\varepsilon}_{\beta} \; = \; \sideset{_{}^{\{\omega \cdot (1 + \beta)\}}}{}{\omega},\;$ which holds for all ordinals $\beta.$

These results and much more -- formulas for the transfinite sequence of fixed points of $\alpha$'th level operations on ordinal numbers (where $\alpha$ is any finite or transfinite ordinal) -- can be found in An extended arithmetic of ordinal numbers by Doner and Tarski (1969; another copy), along with an explanation of why the "bottom-up" formulations for the operations are used (see the first full paragraph on p. 113).