User:RhubarbJayde/REL-NPR
Relativized nonprojectibility, abbreviated REL-NPR, is a systematic extension of a particular characterisation of nonprojectible ordinals. In general, we say \(\alpha\) is a \(\Gamma\)-cardinal iff, for all \(\gamma < \alpha\), there is no surjection \(\pi: \gamma \to \alpha\) with \(\pi \in \Gamma\). This is motivated by the fact that:
- \(\alpha\) is a cardinal iff it is a \(V_{\alpha+1}\)-cardinal.
- \(\alpha\) is a gap iff it is an \(L_{\alpha+1}\)-cardinal.
- \(\alpha\) is \(\Sigma_2\)-admissible iff it is a \(W_{\alpha+2}\)-cardinal.
- \(\alpha\) is nonprojectible iff it is a \(\Sigma_1(L_\alpha)\)-cardinal.
- \(\alpha\) is admissible iff it is a \(W_{\alpha+1}\)-cardinal.
Where \(W_{\omega \alpha+n} = \Delta_n(L_\alpha)\).
Relativized nonprojectibility will be useful in making a "maximal OCF" where one actually manages to collapse practically all recursive structure and \(\omega_1\), e.g. in an ordinal analysis of \(\mathrm{ZFC}^-\) augmented by the existence of \(\omega_1\).
In general, you can assume the existence of \(0^\dagger\) but the nonexistence of \(0^{\dagger \sharp}\), then you're able to collapse \(\omega_1^L\), \(\omega_1^{L[0^\sharp]}\), \(\omega_1^{L[0^{\sharp \sharp}]}\), and then \(\omega_1\) acts as a diagonalizer of \(\alpha \mapsto \omega_\alpha^{L[U]}\). This is super powerful due to the vast order-type of this structure, and leaves many gap-related notions in the dust.