User blog:Leftunknown/integer epsilons

if we use the rule of

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

$$\varepsilon_{\alpha + 1} = \varepsilon_{\alpha}^{\varepsilon_{\alpha}^{\cdots}}$$ we can deduce the following:
 * $$\varepsilon_{-1} = \omega$$
 * $$\varepsilon_{-2} = 1 + \frac{1}{\omega}$$
 * $$\varepsilon_{-3} < \varepsilon_{-2}$$

$$\cdots$$


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

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