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