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Is there a name for the gap between the real numbers and the fractions of $\omega$?

$$\left\{1,2,3,\dotsc\ \middle|\ \omega, \frac{\omega}{2}, \frac{\omega}{3},\dotsc\right\}$$

Is this written about in On Numbers & Games or any other book?

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This is not a "gap" in the technical sense, it's just a regular surreal number. Specifically, it can be written $\sqrt{\omega}$. You can check that that is correct by doing the multiplication.

For a free resource, Equation (5.15) on page 42 of Claus Tøndering's Surreal Numbers - An Introduction states that $$\sqrt{\omega} = \left\{ \mathbb{Z} \ \middle|\ \omega,\frac{\omega}2,\frac{\omega}3,\frac{\omega}4,\dotsc \right\},$$ and the nonpositive numbers on the left don't affect the value by a version of "the simplicity theorem" or a variant of "deleting dominated options" (Theorem 11 on page 21 of Surreal Numbers - An Introduction is enough, for instance.).

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  • $\begingroup$ Thanks for the quick response & link! As a follow up, is there a name for the number $\{\mathbb{z}|\sqrt{\omega},\sqrt[3]{\omega},\sqrt[4]{\omega},...\}$ $\endgroup$ – user820789 Mar 3 '20 at 20:43
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    $\begingroup$ @meowzz, Using the surreal $\omega$-power, I believe that is $\omega^{1/\omega}$. But setting up the background to justify that answer can't be done in a comment, and there may not be free resources with the definition and theorems needed. That might be worth another question all on its own, and if you post one maybe someone will have a resource I don't. $\endgroup$ – Mark S. Mar 3 '20 at 23:34

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