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→A "normal" variant: remove my final sub-example (did not illustrate what I thought it did) |
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The first values of <math>\tilde\psi</math> coincide with those of <math>\psi</math>: namely, for all <math>\alpha<\zeta_0</math> where <math>\zeta_0 = \varphi_2(0)</math>, we have <math>\tilde\psi(\alpha) = \psi(\alpha)</math> because the additional clause <math>\alpha \in \tilde C(\alpha,\rho)</math> is always satisfied. But at this point the functions start to differ: while the function <math>\psi</math> gets “stuck” at <math>\zeta_0</math> for all <math>\zeta_0 \leq \alpha \leq \Omega</math>, the function <math>\tilde\psi</math> satisfies <math>\tilde\psi(\zeta_0) = \varepsilon_{\zeta_0+1}</math> because the new condition <math>\alpha \in \tilde C(\alpha,\rho)</math> imposes <math>\tilde\psi(\zeta_0) > \zeta_0</math>. On the other hand, we still have <math>\tilde\psi(\Omega) = \zeta_0</math> (because <math>\Omega \in C(\alpha,\rho)</math> for all <math>\rho</math> so the extra condition does not come in play). Note in particular that <math>\tilde\psi</math>, unlike <math>\psi</math>, is not monotonic, nor is it continuous.
Despite these changes, the <math>\tilde\psi</math> function also defines a system of ordinal notations up to the Bachmann-Howard ordinal: the notations, and the conditions for canonicalness, are slightly different (for example, <math>\psi(\Omega+1+\alpha) = \tilde\psi(\tilde\psi(\Omega)+\alpha)</math> for all <math>\alpha</math> less than the common value <math>\psi(\Omega2) = \tilde\psi(\
== Collapsing large cardinals ==
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