Proof-theoretic dilator and intermediate pointclasses (2501.11220v1)

Abstract: There are two major generalizations of the standard ordinal analysis: One is Girard's $\Pi^1_2$-proof theory in which dilators are assigned to theories instead of ordinals. The other is Pohlers' generalized ordinal analysis with Spector classes, where ordinals greater than $\omega_1^{\mathsf{CK}}$ are assigned to theories. In this paper, we show that these two are systematically entangled, and $\Sigma^1_2$-proof theoretic analysis has a critical role in connecting these two.

Insightful Overview of the Paper on Proof-theoretic Dilator and Intermediate Pointclasses

The paper by Hanul Jeon, "Proof-theoretic dilator and intermediate pointclasses," addresses the intricate entanglement between two generalizations of ordinal analysis: Girard's $\Pi^1_2$-proof theory involving dilators and Pohlers' generalized ordinal analysis with Spector classes. The work seeks to bridge these frameworks, highlighting the critical role of $\Sigma^1_2$-proof theoretic analysis. The primary focus is on establishing systematic connections between the notions of proof-theoretic dilators and $\Pi^1_1[R]$-proof theoretic ordinals for a $\Sigma^1_2$-singleton real $R$.

The paper is centered around the construction and analysis of genedendrons—structures that manage complexity for $\Sigma^1_2$-singletons by extending concepts of predilators and quasidendroids. The author leverages proof-theoretic methods to define these genedendrons rigorously and uses them to explore $\Sigma^1_2$-proof theoretic analysis. An important technical achievement within is the application of genedendrons to deduce the $\Sigma^1_2$-altitudes—a measure for the complexity of $\Sigma^1_2$-singleton reals—and to illuminate the links to iterated hyperjumps.

By employing a $\Pi^1_1[R]$-proof theoretic framework, the paper systematically intersects analysis involving intermediate pointclasses such as $\Pi^1_1[R]$ with dilator-based proof theory. Jeon effectively uses genedendrons as tools to extract proof-theoretic information about $\Pi^1_1[R]$-consequences from the $\Pi^1_2$ and $\Sigma^1_2$ layers of a theory, which is particularly significant given the layered hierarchy of logics.

The main result demonstrates that, given a $\Pi^1_2$-sound theory $T$ extending $ACA_0$, one can equate elements of the proof-theoretic dilator $|T|_{\Pi^1_2}$ with $\Pi^1_1[R]$-ordinals under specific conditions related to locally well-founded genedendrons. These results provide a formal assurance of the connection between the mentioned proofs and the existence of certain recursive genedendrons, bridging specific levels of the analytical constructs used in their formulation.

Additionally, Jeon highlights a theoretical perspective on the $\Sigma^1_2$-altitude of the hyperjump of $\emptyset$, showing that it corresponds to $\omega_1^CK$, using notions of $-logic. The use of$-logic preproof properties mirrors the role of cut-elimination in other proof-theoretic contexts, hence providing a proof-theoretic analysis through $$-structures similar to those seen in earlier extensions of logic.

The implications of these findings suggest new pathways and frameworks for understanding the logical foundations and computational hierarchies in proof theory. By adopting a parameterized approach to genedendrons for different sections of the proof-theoretic dilator, this research opens up new possibilities for analyzing the complexities within intermediate pointclasses and suggests more nuanced applications in the advancing realms of mathematics and logic theory.

Overall, the paper enriches our comprehension of how ordinal analysis can be methodically linked to different pointclasses, illustrating the versatility and depth by which foundational mathematics and proof theory can model and manipulate complex logical objects like genedendrons and dilators. This work sets the stage for future elaboration on these methods, potentially extending their applications as tools within complexity theory and formal logical systems.