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In this MathOverflow question, the ordinal $\omega\uparrow^\omega\omega$ is defined. It seems like a quite natural ordinal to me, since it is of course \begin{equation} \begin{array}{*8{>{\displaystyle}c}} & \omega \uparrow^\omega \omega \\ = & \sup \{ & \omega \uparrow^0 2, & \omega \uparrow^1 2, & \omega \uparrow^2 2, & \omega \uparrow^3 2, & \ldots & \} \\ = & \sup \{ & \omega \cdot 2 = \omega + \omega, & \omega^2 = \omega \cdot \omega, & ^2 \omega = \omega^\omega, & ^\omega\omega = \omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}} & \ldots & \} \\ \end{array} \end{equation} So in a sense it is the result of saying "and so on" to the game of always getting to the next hyperoperation by saying "and so on".
What is the name of this ordinal?

Smiley1000
  • 950
  • I believe this is the Feferman-Schütte ordinal (see here), although you have to be careful in what $\uparrow^n$ means for $n > 2$ (or perhaps it's for $n > 3),$ as discussed in the comments to Joel David Hamkins's answer that you essentially cited, regarding which see also my comments about weak and strong tetration in this answer, which seem to be similar to Simply Beautiful Art's comment involving up arrows and down arrows. – Dave L. Renfro Aug 10 '22 at 10:05
  • I just realized that most of my discussion about the problems of using the stronger versions of the higher-order operations for (transfinite) ordinals is in my answer to Why are tetrations not useful?, but unfortunately that question was closed and thus probably not viewable without sufficient reputation. One day I'll probably incorporate that discussion (and the mathoverflow answer I previously cited here) into something more extensive, but for now send me an email if you're interested and I'll send screen shots of my closed MSE answer. – Dave L. Renfro Aug 10 '22 at 10:20
  • A similar non-closed question is this https://math.stackexchange.com/questions/583175/, in which it's argued $\varphi_\omega(0)$ is the answer, which is less than Feferman-Schutte. – C7X Aug 25 '22 at 20:28

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