Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-22T20:20:15.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Two further combinatorial theorems equivalent to the 1-consistency of peano arithmetic

Published online by Cambridge University Press:  12 March 2014

Peter Clote  and
Kenneth Mcaloon
Peter Clote
Affiliation:
U.E.R. de Mathématiques, Université Paris VII, 75005 Paris, France
Kenneth Mcaloon
Affiliation:
Brooklyn College of Cuny, Brooklyn, New York 11210
Get access Rights & Permissions [Opens in a new window]

Extract

We give two new finite combinatorial statements which are independent of Peano arithmetic, using the methods of Kirby and Paris [6] and Paris [12]. Both are in fact equivalent over Peano arithmetic (denoted by P) to its 1-consistency. The first involves trees and the second linear orderings. Both were “motivated” by anti-basis theorems of Clote (cf. [1], [2]). The one involving trees, however, is not unrelated to the Kirby-Paris characterization of strong cuts in terms of the tree property [6], but, in fact, comes directly from König's lemma, of which it is a miniaturization. (See the remark preceding Theorem 3 below.) The resulting combinatorial statement is easily seen to imply the independent statement discovered by Mills [11], but it is not clear how to show their equivalence over Peano arithmetic without going through 1-consistency. The one involving linear orderings miniaturizes the property of infinite sets X that any linear ordering of X is isomorphic to ω or ω* on some infinite subset of X. Both statements are analogous to Example 2 of [12] and involve the notion of dense [12] or relatively large [14] finite set.

We adopt the notations and definitions of [6] and [12]. We shall in particular have need of the notions of semiregular, regular and strong initial segments and of indicators.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 4 , December 1983 , pp. 1090 - 1104
Copyright
Copyright © Association for Symbolic Logic 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Clote, P., Anti-basis theorems and their relation to independence results in Peano arithmetic, Model theory and arithmetic, Lecture Notes in Mathematies, vol. 890, Springer-Verlag, Berlin, 1981, pp. 115–133.CrossRefGoogle Scholar
[2]Clote, P., A note on decidable models, Model theory and arithmetic, Lecture Notes in Mathematics, vol. 890, Springer-Verlag, Berlin, 1981, pp. 134–142.CrossRefGoogle Scholar
[3]Friedman, H., Systems of second order arithmetic with restricted induction. I, II (abstract), this Journal, vol. 41 (1976), pp. 557–558, 558–559.Google Scholar
[4]Friedman, H., Some systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematicians (Vancouver, 1974), Vol. 1, Canadian Mathematical Congress, Montréal, 1975, pp. 235–242.Google Scholar
[5]Ketonen, J. and Solovay, R., Rapidly growing Ramsey functions, Annals of Mathematics, ser. 2, vol. 113 (1981), pp. 267–314.CrossRefGoogle Scholar
[6]Kirby, L.A.S. and Paris, J.B., Initial segments of models of Peano's axioms, Set theory and hierarchy theory. V, Lecture Notes in Mathematics, vol. 619, Springer-Verlag, Berlin, 1977, pp. 211–226.CrossRefGoogle Scholar
[7]Kirby, L.A.S., Initial segments of models of arithmetic, Ph. D. Thesis, University of Manchester, Manchester, 1977.Google Scholar
[8]Kirby, L.A.S., Flipping properties in arithmetic, this Journal, vol. 47 (1982), pp. 416–422.Google Scholar
[9]MacIntyre, A., Ramsey quantifiers in arithmetic, Model theory of algebra and arithmetic, Lecture Notes in Mathematics, vol. 834, Springer-Verlag, Berlin, 1980, pp. 186–210.CrossRefGoogle Scholar
[10]McAloon, K., Les rapports entre la méthode des indicatrices et la méthode de Gödel pour obtenir des résultats d'indépendance, Modèles de l'arithmétique, Astérisque No 73, Société Mathématique de France, Paris, 1980, pp. 31–39.Google Scholar
[11]Mills, G., A tree analysis of unprovable combinatorial statements, Model theory of algebra and arithmetic, Lecture Notes in Mathematics, vol. 834, Springer-Verlag, Berlin, 1980, pp. 248–311.CrossRefGoogle Scholar
[12]Paris, J.B., Some independence results for Peano arithmetic, this Journal, vol. 43 (1978), pp. 725–731.Google Scholar
[13]Paris, J.B., A hierarchy of cuts in models of arithmetic, Model theory of algebra and arithmetic, Lecture Notes in Mathematics, vol. 834, Springer-Verlag, Berlin, 1980, pp. 312–337.CrossRefGoogle Scholar
[14]Paris, J.B. and Harrington, L., A mathematical incompleteness in Peano arithmetic, Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 1133–1142.CrossRefGoogle Scholar