List of ordinals: Difference between revisions

The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for <math>\(\alpha<\omega_1^{M_n}</math>\), <math>\(\alpha</math>\) is <math>\(M_n</math>\)-stable iff it is <math>\(\Sigma^1_{n+2}</math>\)-reflecting when <math>\(n</math>\) is even, and <math>\(\Pi^1_{n+2}</math>\)-reflecting when <math>\(n</math>\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19-->