Thread:Leftunknown/@comment-36856644-20191019124032

if we use the rule of

$$\varepsilon_{\alpha + 1} = min{\beta|\beta = \omega^{\beta} \and \beta > \varepsilon_{\alpha}}$$ , which can also be

$$\varepsilon_{\alpha + 1} = \varepsilon_{\alpha}^{\varepsilon_{\alpha}^{\cdots}}$$ we can deduce the following:

$$\varepsilon_{-1} = \omega$$

$$\varepsilon_{-2} = 1 + \frac{1}{\omega}$$

$$\varepsilon_{-3} < \varepsilon_{-2} \omega$$

$$\cdots$$

$$\varepsilon_{\varepsilon_{-1}} = \varepsilon_{\omega}$$

$$\varepsilon_{\varepsilon_{-2}} = \varepsilon_{1}$$

$$\varepsilon_{-\varepsilon_{-1}} = \varepsilon_{-\omega}$$

$$\varepsilon_{-\varepsilon_{-2}} = \omega$$

although, set theory doesn't work with integers, so im not sure how accurate this is 