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Simple formulas defining complicated sets
Author(s) -
Lücke Philipp
Publication year - 2016
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201610442
Subject(s) - simple (philosophy) , axiom , mathematics , combinatorics , mathematical economics , discrete mathematics , philosophy , epistemology , geometry
We consider the question whether large cardinal axioms imply that certain complicated sets cannot be defined by simple formulas. More precisely, we ask whether the existence of larger large cardinals is compatible with the existence of a well‐ordering of the real numbers that is definable by a Σ 1 ‐formula that uses a single ordinal as a parameter. This note presents results by Ralf Schindler, Philipp Schlicht and the author showing that the existence of a well‐ordering of the reals that is definable by a Σ 1 ‐formula with parameter ω 1 is compatible with the existence of a Woodin cardinal and incompatible with the existence of a Woodin cardinal with a measurable cardinal above it. Moreover, a similar result holds for Σ 1 ‐formulas using certain large cardinals as a parameter. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)