User:Username5243/Pi notation p2

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Extended pi notation
Last page, we left off at an impass; we reached the ordinal \(\psi(\Omega_\omega)\), which happens to be equal to \(\pi(\psi(\Omega_\omega))\). It would be nice to develop a nontation for such fixed poins.

Which is where the \(\Omega\) comes in. If you find a \(\Omega\) in the first argument, you get the function that it is a fixed point of by replacing the last \(\Omega\) by a \(\alpha\) \(\pi(\Omega,\alpha)\) gives fixed points of \(\alpha = \pi(\alpha)\). Then we get \(\pi(\Omega+1), \pi(\Omega+\omega), \pi(\Omega+\omega[\alpha], \(\pi(\omega+\alpha)\)... And the limit of all that is \(\pi(\Omega2)\), and \(\pi(\Omega2,\alpha)\) enumerates fixed points of \(\alpha = \pi(\Omega+\alpha)\). Then \(\pi(\Omega3), \pi(\Omega\omega), \pi(\Omega^2), \pi(\Omega^\Omega), \pi(\varepsilon_{\Omega+1}\)...

This notation will be able to take us a VERY long way - even \(\pi(\Omega2)\) is a huge ordinal (based on my preliminary analysis I think it goes far beyond weakly compact cardinals). Let's get going...