The Copernican Multiverse of Sets
| Autor: | Paul K. Gorbow, Graham E. Leigh |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Logic
Absoluteness media_common.quotation_subject Provability logic Mathematics - Logic Copernican principle Universe Physics::History of Physics Philosophy symbols.namesake Range (mathematics) Theoretical physics Mathematics (miscellaneous) Multiverse 03A05 03E30 03E35 03E65 03H05 symbols FOS: Mathematics Set theory Logic (math.LO) Axiom Mathematics media_common |
| ISSN: | 1755-0203 |
| Popis: | We develop an untyped framework for the multiverse of set theory.$\mathsf {ZF}$is extended with semantically motivated axioms utilizing the new symbols$\mathsf {Uni}(\mathcal {U})$and$\mathsf {Mod}(\mathcal {U, \sigma })$, expressing that$\mathcal {U}$is a universe and that$\sigma $is true in the universe$\mathcal {U}$, respectively. Here$\sigma $ranges over the augmented language, leading to liar-style phenomena that are analyzed. The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as aCopernican principlethat the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory$\mathsf {ZF}$, and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins’ multiverse theory. |
| Databáze: | OpenAIRE |
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