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Rank-initial embeddings of non-standard models of set theory

Artikel i vetenskaplig tidskrift
Författare Paul Kindvall Gorbow
Publicerad i Archive for Mathematical Logic
ISSN 0933-5846
Publiceringsår 2019
Publicerad vid Institutionen för filosofi, lingvistik och vetenskapsteori
Språk en
Länkar https://doi.org/10.1007/s00153-019-...
Ämnesord Automorphism, Embedding, Fixed point, GBC, Iterated ultrapower, KP, Nonstandard model, Recursively saturated, Self-embedding, Set theory, Strong cut, Weakly compact, ZF
Ämneskategorier Matematisk logik

Sammanfattning

A theoretical development is carried out to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman’s theorem on the existence of rank-initial embeddings between countable non-standard models of a fragment of Zermelo–Fraenkel set theory (ZF); and Gaifman’s technique of iterated ultrapowers is employed to show that any countable model of GBC + ``Ord is weakly compact'' (where GBC is Gödel-Bernays set theory with choice) can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of “strong rank-cut” is characterized (i) in terms of the theory GBC + ``Ord is weakly compact'', and (ii) in terms of fixed-point sets of self-embeddings.

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