Journal of Symbolic Logic 75 (4):1311-1325 (2010)

Abstract
An open U ⊆ ℝ is produced such that (ℝ, +, ·, U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ, +, ·). In particular, there is a Cantor set E ⊆ ℝ such that (ℝ, +, ·, E) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansion $\germ{R}$ of (ℝ, +, ·), every subset of ℝ definable in ( $\germ{R}$ , E) either has interior or is Hausdorff null
Keywords expansion of the real field   o-minimal   projective hierarchy   Cantor set   Hausdorff dimension   Minkowski dimension
Categories (categorize this paper)
DOI 10.2178/jsl/1286198148
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 54,385
Through your library

References found in this work BETA

Expansions of o-Minimal Structures by Iteration Sequences.Chris Miller & James Tyne - 2006 - Notre Dame Journal of Formal Logic 47 (1):93-99.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

Expansions Which Introduce No New Open Sets.Gareth Boxall & Philipp Hieromyni - 2012 - Journal of Symbolic Logic 77 (1):111-121.
What is o-Minimality?Harvey M. Friedman - 2008 - Annals of Pure and Applied Logic 156 (1):59-67.
Interpretability Over Peano Arithmetic.Claes Strannegård - 1999 - Journal of Symbolic Logic 64 (4):1407-1425.
Presburger Sets and P-Minimal Fields.Raf Cluckers - 2003 - Journal of Symbolic Logic 68 (1):153-162.
Expansions of o-Minimal Structures by Fast Sequences.Harvey Friedman & Chris Miller - 2005 - Journal of Symbolic Logic 70 (2):410-418.
Definable Open Sets As Finite Unions of Definable Open Cells.Simon Andrews - 2010 - Notre Dame Journal of Formal Logic 51 (2):247-251.

Analytics

Added to PP index
2010-09-12

Total views
16 ( #603,641 of 2,362,053 )

Recent downloads (6 months)
1 ( #553,136 of 2,362,053 )

How can I increase my downloads?

Downloads

My notes