Bootstraps, nets, and hierarchies

摘要: Classification is at the heart of scientific enterprise, from bacteria in biology to groups mathematics. A central classification project mathematical logic motivated by Goedel's incompleteness theorems. Indeed, logical systems are classified according how hard it establish that no contradiction can be derived these systems, yielding Goedel hierarchy: a linear hierarchy claimed encompass all natural/foundationally important systems. The medium range this based on second-order arithmetic, system with roots Hilbert-Bernays' Grundlagen der Mathematik. exhibits remarkable robustness: enriching language higher-order arithmetic does not change picture, while switching inclusion ordering introduces only few natural outliers, and also parallel for axiom determinacy set theory. In paper, we introduce numerous such hierarchies inclusion-based hierarchy, basic convergence theorems nets. Among hierarchies, identify Plato which yields (medium of) under canonical embedding into arithmetic. defined Grundlagen, erstwhile preserves equivalences, translating e.g. an equivalence involving monotone theorem nets well-known arithmetical comprehension sequences.