Localizing the axioms
| Autor: | Athanassios Tzouvaras |
|---|---|
| Rok vydání: | 2023 |
| Předmět: |
Discrete mathematics
Morse–Kelley set theory Logic Internal set theory Mathematics::General Topology Mathematics - Logic Inner model theory New Foundations Mathematics::Logic Philosophy symbols.namesake Large cardinal Von Neumann–Bernays–Gödel set theory Rank-into-rank Inaccessible cardinal symbols FOS: Mathematics Logic (math.LO) Mathematics |
| DOI: | 10.48550/arxiv.2303.15264 |
| Popis: | We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by $Loc({\rm ZFC})$, says that every set belongs to a transitive model of ZFC. LZFC consists of $Loc({\rm ZFC})$ plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All $\Pi_2$ consequences of ZFC are provable in ${\rm LZFC}$. LZFC strongly extends Kripke-Platek (KP) set theory minus $\Delta_0$-Collection and minus $\in$-induction scheme. ZFC+``there is an inaccessible cardinal'' proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define $\alpha$-Mahlo models and $\Pi_1^1$-indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form $Loc({\rm ZFC}+\phi)$ are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved. Comment: 38 pages |
| Databáze: | OpenAIRE |
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