Oneiric Numbers

Note that $\uparrow$ is not a number: it is the value of a game, which is a more subtle concept. Also note that $\frac 1 \uparrow$ is not defined since it would be bigger than all surreal numbers and there are no such numbers. (In fact, it does exist but is one of the Oneiric numbers.)

More Infinite Games - Conway


$$+_0=\{0||0|0\}=\{0|*\}=\uparrow$$ $$+_{\text{on}}=\{0||0|\text{off}\}=\text{pip}_0=\text{tiny}$$ $$\text{over}=\frac 1 {\text{on}}$$

Consider an inversion such that

$$\text{on}=\frac 1 {\text{over}}$$ $$\frac 1 \uparrow=\frac 1 +_0=I$$ $$\frac 1 +_{\text{on}}=\frac 1 {\text{tiny}}=\text{huge}$$

Define $\mathbb{O}\text{ne}$(iric #s) as the domain of games from $\text{massive}(-\text{huge}) \rightarrow \text{huge}$:


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How do $\text{over}$ & $\frac 1 {\text{on}}$ relate to $+_{\text{over}}$ & $+_{\frac 1 {\text{on}}}$?

What is the value of $\text{pip}_\text{on}$? $\uparrow$? $\text{over}$? Or..?

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