Source: J. Symbolic Logic
Volume 73, Issue 3
Some 40 years ago, Dana Scott proved that every countable Scott set
is the standard system of a model of PA. Two decades later, Knight
and Nadel extended his result to Scott sets of size ω1.
Here, I show that assuming the Proper Forcing Axiom (PFA), every
A-proper Scott set is the standard system of a model of PA. I
define that a Scott set 𝔛 is proper if the quotient Boolean
algebra 𝔛/Fin is a proper partial order and A-proper if 𝔛 is
additionally arithmetically closed. I also investigate the question
of the existence of proper Scott sets.
J. E. Baumgartner, Applications of the proper forcing axiom, Handbook of set theoretic topology (K. Kunen and J. Vaughan, editors), North--Holland, Amsterdam, 1984, pp. 913--959.
Mathematical Reviews (MathSciNet): MR776640
A. Blass, On certain types and models for arithmetic, Journal of Symbolic Logic, vol. 39 (1974), pp. 151--162.
Mathematical Reviews (MathSciNet): MR369050
A. Enayat, From bounded arithmetic to second order arithmetic via automorphisms, Logic in Tehran (A. Enayat, I. Kalantari, and M. Moniri, editors), Lecture Notes in Logic, vol. 26, Association for Symbolic Logic, 2006, pp. 87--113.
A. Enayat, Uncountable expansions of the standard model of Peano arithmetic, preprint, 2006.
F. Engström, Expansions, omitting types, and standard systems, Ph.D. thesis, Chalmers University of Technology and Goteborg University, 2004.
H. Gaifman, Models and types of Peano's arithmetic, Annals of Mathematical Logic, vol. 9 (1976), no. 3, pp. 223--306.
Mathematical Reviews (MathSciNet): MR406791
V. Gitman, Applications of the Proper Forcing Axiom to models of Peano Arithmetic, Ph.D. thesis, The Graduate Center of the City University of New York, 2007.
V. Gitman, Proper and piecewise proper families of reals, 2007, preprint.
T. Jech, Set theory, third ed., Springer Monographs in Mathematics, Springer--Verlag, New York, 2003.
T. Johnstone and J. D. Hamkins, The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph_2$ or $\aleph_3$, (2007), preprint.
R. Kaye, Models of Peano arithmetic, Oxford Logic Guides, vol. 15, Oxford University Press, New York, 1991.
L. Kirby and J. B. Paris, Initial segments of models of Peano's axioms, Set theory and hierarchy theory, V (Proceedings of the third conference, Bierutowice, 1976), Lecture Notes in Mathematics, vol. 619, Springer--Verlag, Berlin, 1977, pp. 211--226.
Mathematical Reviews (MathSciNet): MR491157
J. Knight and M. Nadel, Models of Peano arithmetic and closed ideals, Journal of Symbolic Logic, vol. 47 (1982), no. 4, pp. 833--840.
Mathematical Reviews (MathSciNet): MR683158
R. Kossak and J. H. Schmerl, The structure of models of Peano arithmetic, Oxford Logic Guides, vol. 50, Oxford University Press, New York, 2006.
D. Scott, Algebras of sets binumerable in complete extensions of arithmetic, Proceedings of the symposium on pure mathematics vol. V, American Mathematical Society, Providence, R.I., 1962, pp. 117--121.
Mathematical Reviews (MathSciNet): MR141595
S. Shelah, Proper and improper forcing, second ed., Perspectives in Mathematical Logic, Springer--Verlag, New York, 1998.
S. Smoryński, Lectures on nonstandard models of arithmetic, Logic colloquium '82 (Florence, 1982), Studies in Logic and the Foundations of Mathematics, vol. 112, North--Holland, Amsterdam, 1984, pp. 1--70.
Mathematical Reviews (MathSciNet): MR762103
B. Veličković, Forcing axioms and stationary sets, Advances in Mathematics, vol. 94 (1992), no. 2, pp. 256--284.
A. Villaveces, Chains of end elementary extensions of models of set theory, Journal of Symbolic Logic, vol. 63 (1998), no. 3, pp. 1116--1136.