Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Peano arithmetic and hyper-Ramsey logic

Author: James H. Schmerl
Journal: Trans. Amer. Math. Soc. 296 (1986), 481-505
MSC: Primary 03H15; Secondary 03C80, 03C85, 03F35
MathSciNet review: 846594
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is known that $ {\text{PA}}({Q^2})$, Peano arithmetic in a language with the Ramsey quantifier, is complete and compact and that its first-order consequences are the same as those of $ \Pi _1^1{\text{-CA}_0}$. A logic $ \mathcal{H}{\mathcal{R}_\omega }$, called hyper-Ramsey logic, is defined; it is the union of an increasing sequence $ \mathcal{H}{\mathcal{R}_1} \subseteq {\mathcal{H}_{\mathcal{R}2}} \subseteq \mathcal{H}{\mathcal{R}_3} \subseteq \cdots $ of sublogics, and $ \mathcal{H}{\mathcal{R}_1}$ contains $ L({Q^2})$. It is proved that $ {\text{PA}}(\mathcal{H}{\mathcal{R}_n})$, which is Peano arithmetic in the context of $ \mathcal{H}{\mathcal{R}_n}$, has the same first-order consequences as $ \Pi _n^1{\text{-CA}_0}$. A by-product and ingredient of the proof is, for example, the existence of a model of $ {\text{CA}}$ having the form $ (\mathcal{N}, {\text{Class}}(\mathcal{N}))$.

References [Enhancements On Off] (What's this?)

  • [1] Matt Kaufmann, A rather classless model, Proc. Amer. Math. Soc. 62 (1977), no. 2, 330–333. MR 0476498, 10.1090/S0002-9939-1977-0476498-7
  • [2] Handbook of mathematical logic, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Edited by Jon Barwise; With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra; Studies in Logic and the Foundations of Mathematics, Vol. 90. MR 0457132
  • [3] R. Mac Dowell and E. Specker, Modelle der Arithmetik, Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959), Pergamon, Oxford; Państwowe Wydawnictwo Naukowe, Warsaw, 1961, pp. 257–263 (German). MR 0152447
  • [4] Angus Macintyre, Ramsey quantifiers in arithmetic, Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), Lecture Notes in Math., vol. 834, Springer, Berlin-New York, 1980, pp. 186–210. MR 606787
  • [5] Menachem Magidor and Jerome Malitz, Compact extensions of 𝐿(𝑄). Ia, Ann. Math. Logic 11 (1977), no. 2, 217–261. MR 0453484
  • [6] James H. Schmerl, Peano models with many generic classes, Pacific J. Math. 46 (1973), 523–536. MR 0354351
  • [7] James H. Schmerl, Recursively saturated, rather classless models of Peano arithmetic, Logic Year 1979–80 (Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, Conn., 1979/80) Lecture Notes in Math., vol. 859, Springer, Berlin-New York, 1981, pp. 268–282. MR 619874
  • [8] James H. Schmerl and Stephen G. Simpson, On the role of Ramsey quantifiers in first order arithmetic, J. Symbolic Logic 47 (1982), no. 2, 423–435. MR 654798, 10.2307/2273152
  • [9] J. H. Schmerl, Peano arithmetic and hyper-Ramsey logic, Abstracts Amer. Math. Soc. 3 (1982), 412.
  • [10] W. Sieg, Conservation theorems for subsystems of analysis with restricted induction (abstract), J. Symbolic Logic 46 (1981), 194.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03H15, 03C80, 03C85, 03F35

Retrieve articles in all journals with MSC: 03H15, 03C80, 03C85, 03F35

Additional Information

Keywords: Peano arithmetic, second-order theories of arithmetic, Ramsey quantifier, ramified analytical hierarchy, hyper-Ramsey logic
Article copyright: © Copyright 1986 American Mathematical Society