Proper and piecewise proper families of reals

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 @ARTICLE {gitman:proper, AUTHOR = {Victoria Gitman}, TITLE = {Proper and Piecewise Proper Families of Reals}, JOURNAL = {Mathematical Logic Quarterly}, VOLUME = {55}, YEAR = {2009}, NUMBER = {5}, PAGES = {542-550}, EPRINT ={0801.4368}, ISSN = {0942-5616}, MRCLASS = {03E35 (03E40 03H15)}, MRNUMBER = {2568765 (2011c:03109)}, MRREVIEWER = {Renling Jin}, DOI = {10.1002/malq.200810015}, URL = {http://dx.doi.org/10.1002/malq.200810015}, }

V. Gitman, “Proper and Piecewise Proper Families of Reals,” Mathematical Logic Quarterly, vol. 55, no. 5, pp. 542–550, 2009.

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 [1]. 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 $\leq\omega_1$.

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 ${\rm ZFC}$ to have continuum many proper families of cardinality $\omega_1$ and continuum many piecewise proper families of cardinality $\omega_2$.

References

1. V. Gitman, “Scott’s problem for proper Scott sets,” J. Symbolic Logic, vol. 73, no. 3, pp. 845–860, 2008.