User:Vel!/pu/Hyperdiagonalization

Say you have a series of ordinals. You want to make sure all of them are 'notable', and all recursive. So you go:

$$\omega, \varepsilon_0, \Gamma_0, \psi(\Omega^{\Omega^{\omega}}), \psi(\Omega^{\Omega^{\Omega}}), \psi(\varepsilon_{\Omega+1}), \psi(\Omega_{\omega}), \psi(\psi_{I}(0)), ...$$

But what comes next? $$\psi(\psi_{I}(I))$$? $$\psi(\psi_{I_{2}}(0))$$? $$\psi(\psi_{\chi(M)}(0))$$? $$\psi(\psi_{\chi(\varepsilon_{M+1})}(0))$$? $$\psi(\psi_{M}(0))$$? Are all those well defined??

We need a series that constantly diagonalizes over itself. If there is an available higher-order diagonalization, then diagonalize whenever possible. Don't let monotony break out. Those are the rules for hyperdiagonalization, but how do you formalize them?

Say you did. Now, what is the limit? Is it $$\omega_{1}^{CK}$$? No. This is recursion, you cannot define $$\omega_{1}^{CK}$$ with recursive methods. This seems to be a new sort of ordinal, pushing "recursive" to its limit: the Sam Ordinals (named after Sam's Number, a "number" that isn't even defined). They are only very loosely defined, and their existence depends on the well-definedness of hyperdiagonalization.

Sam ordinals are so big that infinitely many notable ordinals can be defined before them! Infinitely many new recursive methods can be defined, not forming a monotonous series but continually escalating higher through the recursive ordinals. The non-Sam ordinals, to be specific.

The limit of a hyperdiagonalization of non-Sam ordinals is the first Sam ordinal, $$\omega_{1}^{S}$$. This ordinal and higher recursive ordinals this are called Sam ordinals. They are still technically recursive, but push it to the limits.

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