User blog:Edwin Shade/Negatively Indexed Epsilon Numbers

When we speak of the epsilon-numbers, we speak of those of the form \(\epsilon_{\alpha}\), where \(\alpha\geq 0\). But taking the general principle that if \(\gamma^{\gamma^{\gamma^{\gamma^{.^{.^{.}}}}}}=\epsilon_{\delta}\), where \(\epsilon_{\delta-1}=\gamma\) and \(\epsilon_0=\omega^{\omega^{\omega^{\omega^{.^{.^{.}}}}}}\), it is quite plain to see that \(\epsilon_{-1}\) must equal \(\omega\). By considering \(\omega\) as a number of the form \(\gamma\uparrow\uparrow\omega\), it is also quite clear that \gamma must be a positive finite number greater than 1, and hence \(\epsilon_{-2}\) may assume any positive integer value provided it is within the bounds \(1<n<\omega\).

The calculation of \(\epsilon_{-3}\) is only possible by expressing a number in the aforementioned bounds in the form \(\gamma\uparrow\uparrow\omega\), and as this is only possible with values of n greater than 1 yet \(\seq e\), where e refers to the mathematical constant; and n must be an integer, (for simplicity's sake, though you may assume a non-integer value if you wish), then n may only be 2, and hence \(\epsilon_{-3}\) is the value \(\gamma\) such that \(\gamma\uparrow\uparrow\omega=2\). Here \(\gamma\) is equal to \(\sqrt{2}\), and so \(\epsilon_{-3}=\sqrt{2}\). To compute \(\epsilon_{-4}\) just find the number \(\gamma\) such that \(\gamma\uparrow\uparrow\omega=\epsilon_{-3}\), or \(\sqrt[\sqrt{2}]{\sqrt{2}}\). For the negatively indexed epsilon-number \(\epsilon_{\delta-1}\), it is equivalent to \(\gamma\) where \(\gamma\uparrow\uparrow\omega=\epsilon_{\delta}\). \(\epsilon_{-\omega}\) is therefore equal to the fixed point of \(x\mapsto\sqrt[x]{x}\), or 1. For recessive, (the term "successive" is not appropriate in a situation in which we are enumerating backwards), epsilon-numbers such as \(\epsilon_{-omega-1}\) or \(\epsilon_{{-omega}2}\), they are all equal to 1, since by transfinite induction, if \(\gamma\uparrow\uparrow\omega=1\), then \(\gamma=1\), (therefore implying \(\forall\alpha>\omega\), \(\epsilon_{-\alpha}=\epsilon_{-\beta}\), where \(\beta>\alpha\)), and therefore 1 represents a fixed point of the negatively indexed epsilon numbers.

This has been a brief foray into an extension of the epsilon system inspired by a comment on the Cantor's ordinal talk page. If you have any additional thoughts, questions, or comments, please leave them below.