The strong reflecting property and Harrington's Principle
Yong Cheng
TL;DR
This work analyzes the strong reflecting property for $L$-cardinals and Harrington's Principle $HP(L)$, introducing a generalized principle $HP(M)$ for arbitrary inner models. It systematically develops the theory of $SRP^{L}$, characterizes $SRP^{L}(\omega_n)$ for all $n$, and connects these reflection principles to $0^{\sharp}$ and $ abla_{\,\omega_n}$; it also studies the extension to $SRP^{M}(\gamma)$ and related inner-model phenomena. The paper then maps out the implications and limitations of Harrington-type principles across several inner models, showing equivalences such as $HP(L) ightleftharpoons 0^{\sharp}$ under suitable contexts and identifying cases where $HP(M)$ fails (e.g., core models) or corresponds to $0^{\dag}$ for $L[U]$. Finally, it establishes forcing-based routes to consistency results, demonstrating how $SRP^{L}(\omega_1)$ yields models of $Z_2+HP(L)$ and clarifying the relative strength of $SRP^{L}(\omega_1)$ versus $SRP^{L}(\omega_2)$ under various forcing regimes.
Abstract
In this paper we characterize the strong reflecting property for $L$-cardinals for all $ω_n$, characterize Harrington's Principle $HP(L)$ and its generalization and discuss the relationship between the strong reflecting property for $L$-cardinals and Harrington's Principle $HP(L)$.