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Proper and piecewise proper families of reals
Author(s) -
Gitman Victoria
Publication year - 2009
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200810015
Subject(s) - mathematics , cardinality (data modeling) , piecewise , modulo , partially ordered set , family of sets , quotient , discrete mathematics , ideal (ethics) , nowhere dense set , countable set , cardinal number (linguistics) , peano axioms , combinatorics , pure mathematics , set (abstract data type) , mathematical analysis , philosophy , epistemology , computer science , programming language , linguistics , data mining
I introduced the notions of proper and piecewise proper families of reals to make progress on a long standing open question in the field of models of Peano Arithmetic [5]. A family of reals is proper if it is arithmetically closed and its quotient Boolean algebra modulo the ideal of finite sets is a proper poset. A family of reals is piecewise proper if it is the union of a chain of proper families each of whom has size ≤ ω 1 . Here, I investigate the question of the existence of proper and piecewise proper families of reals of different cardinalities. I show that it is consistent relative to ZFC to have continuum many proper families of cardinality ω 1 and continuum many piecewise proper families of cardinality ω 2 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)