FANDOM


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

Community content is available under CC-BY-SA unless otherwise noted.