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
DOI: https://doi.org/10.1090/S0002-9947-1986-0846594-0
MathSciNet review: 846594
Full-text PDF

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] M. Kaufmann, A rather classless model, Proc. Amer. Math. Soc. 62 (1977), 330-333. MR 0476498 (57:16058)
  • [2] K. Kunen, Combinatorics, Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 371-401. MR 0457132 (56:15351)
  • [3] R. MacDowell and E. Specker, Modelle der Arithmetik, Infinitistic Methods, Proc. Sympos. Foundations of Math. (Warsaw, 1959), Pergamon Press, New York, 1961, pp. 257-263. MR 0152447 (27:2425)
  • [4] A. Macintyre, Ramsey quantifiers in arithmetic, Model Theory of Algebra and Arithmetic, Lecture Notes in Math., vol. 834, Springer-Verlag, Berlin and New York, 1980, pp. 186-210. MR 606787 (83j:03099)
  • [5] M. Magidor and J. I. Malitz, Compact extensions of $ {\text{L}}({\text{Q}})$ (Part 1a), Ann. Math. Logic 11 (1977), 217-261. MR 0453484 (56:11746)
  • [6] J. H. Schmerl, Peano models with many generic classes, Pacific J. Math. 46 (1973), 523-536; Correction 92 (1981), 195-198. MR 0354351 (50:6831)
  • [7] -, Recursively saturated, rather classless models of Peano arithmetic, Logic Year 1979-1980, Lecture Notes in Math., vol. 859, Springer-Verlag, Berlin and New York, 1981, pp. 268-282. MR 619874 (83b:03039)
  • [8] J. H. Schmerl and S. G. Simpson, On the role of the Ramsey quantifiers in first order arithmetic, J. Symbolic Logic 47 (1982), 423-435. MR 654798 (83j:03062)
  • [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

DOI: https://doi.org/10.1090/S0002-9947-1986-0846594-0
Keywords: Peano arithmetic, second-order theories of arithmetic, Ramsey quantifier, ramified analytical hierarchy, hyper-Ramsey logic
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society