Definability of Satisfaction in Outer Models

In Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality. Basel, Switzerland: Birkhäuser. pp. 135-160 (2018)
  Copy   BIBTEX

Abstract

Let M be a transitive model of ZFC. We say that a transitive model of ZFC, N, is an outer model of M if M ⊆ N and ORD ∩ M = ORD ∩ N. The outer model theory of M is the collection of all formulas with parameters from M which hold in all outer models of M. Satisfaction defined with respect to outer models can be seen as a useful strengthening of first-order logic. Starting from an inaccessible cardinal κ, we show that it is consistent to have a transitive model M of ZFC of size κ in which the outer model theory is lightface definable, and moreover M satisfies V = HOD. The proof combines the infinitary logic L∞,ω, Barwise’s results on admissible sets, and a new forcing iteration of length strictly less than κ+ which manipulates the continuum function on certain regular cardinals below κ. In the Appendix, we review some unpublished results of Mack Stanley which are directly related to our topic.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 93,127

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Definability of satisfaction in outer models.Sy-David Friedman & Radek Honzik - 2016 - Journal of Symbolic Logic 81 (3):1047-1068.
The largest countable inductive set is a mouse set.Mitch Rudominer - 1999 - Journal of Symbolic Logic 64 (2):443-459.
Power-like models of set theory.Ali Enayat - 2001 - Journal of Symbolic Logic 66 (4):1766-1782.
On models with power-like ordering.Saharon Shelah - 1972 - Journal of Symbolic Logic 37 (2):247-267.
Hyperclass Forcing in Morse-Kelley Class Theory.Carolin Antos & Sy-David Friedman - 2018 - In Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality. Basel, Switzerland: Birkhäuser. pp. 17-46.
Permutation Models and SVC.Eric J. Hall - 2007 - Notre Dame Journal of Formal Logic 48 (2):229-235.
Downward transfer of satisfiability for sentences of L 1,1.Jerome Malitz - 1983 - Journal of Symbolic Logic 48 (4):1146-1150.
Embeddings Into Outer Models.Monroe Eskew & Sy-David Friedman - 2022 - Journal of Symbolic Logic 87 (4):1301-1321.
Generic coding with help and amalgamation failure.Sy-David Friedman & Dan Hathaway - 2021 - Journal of Symbolic Logic 86 (4):1385-1395.

Analytics

Added to PP
2020-06-17

Downloads
11 (#1,167,245)

6 months
7 (#491,177)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Citations of this work

No citations found.

Add more citations

References found in this work

No references found.

Add more references