Proper and piecewise proper families of reals

✍ Scribed by Victoria Gitman

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 136 KB
• Volume: 55
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200810015

✦ Synopsis

Abstract

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)