Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T05:19:06.153Z Has data issue: false hasContentIssue false
  • English
  • Français

Undefinability of κ-well-orderings in L∞κ

Published online by Cambridge University Press:  12 March 2014

Juha Oikkonen
Juha Oikkonen*
Affiliation:
Helsingin Yliopisto, Matematiikan Laiton, PI 4 (Yliopistonkatu 5), 00014 Helsingin Yliopisto, Finland, E-mail: oikkonen@cc.helsinki.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the class of trees with no branches of cardinality ≤ κ is not RPC definable in L∞κ when κ is regular. Earlier such a result was known for under the assumption κ<κ = κ. Our main result is actually proved in a stronger form which covers also L∞κ (and makes sense there) for every strong limit cardinal λ < κ of cofinality κ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 999 - 1020
Copyright
Copyright © Association for Symbolic Logic 1997

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]Chang, C. C., Two interpolation theorems, Instituto Nazionale di Alta Matematica, Symposia Matematica, vol. V (1970), pp. 5–19.Google Scholar
[2]Cunningham, E., Applications of consistency properties and back-and-forth techniques in infinite-quantifier languages, in Karp-memoirs.Google Scholar
[3]Dickmann, M., Large infinitary languages—model theory, North-Holland, Amsterdam, 1975.Google Scholar
[4]Green, J., Consistency properties for finite quantifier languages, Infinitary logic: In memoriam Carol Karp (Kueker, D. W., editor), Lecture Notes in Mathematics, vol. 492, 1975, pp. 74–123.Google Scholar
[5]Hintikka, J. and Rantala, V., A new approach to infinitary languages, Annals of Mathematical Logic, vol. 10 (1976), pp. 95–115.CrossRefGoogle Scholar
[6]Hyttinen, T., Model theory for infinite quantifier languages, Fundamenta Mathematicae, vol. 134 (1990), pp. 125–142.CrossRefGoogle Scholar
[7]Karp, C., Infinite quantifier languages and ω-chains of models, Proceedings of the Tarski symposium, American Mathematical Society, 1974, pp. 225–232.CrossRefGoogle Scholar
[8]Karttunen, M., Model theory for infinitely deep languages, Annales Academiae Scientiarum Fennicae. Series A I. Mathematica Dissertationes, vol. 50 (1984).Google Scholar
[9]Keisler, H. J., Model theory for infinitary logic, North-Holland, 1971.Google Scholar
[10]López-Escobar, E. G. K., On defining well-orderings, Fundamenta Mathematicae, vol. LIX (1966), pp. 13–21.CrossRefGoogle Scholar
[11]López-Escobar, E. G. K., An addition to On defining well-orderings, Fundamenta Mathematicae, vol. LIX (1966), pp. 299–300.CrossRefGoogle Scholar
[12]Makkai, M., Generalizing Vaught sentences from ω to strong cofinality ω, Fundamenta Mathematicae, vol. LXXXII (1974), pp. 105–119.CrossRefGoogle Scholar
[13]Makkai, M., Admissible sets and infinitary logic, Handbook of mathematical logic (Barwise, J., editor), North-Holland, 1977, pp. 739–782.Google Scholar
[14]Mekler, A. and Väänänen, J., Trees and -subsets of , this Journal, vol. 58 (1993), pp. 1052–1070.Google Scholar
[15]Oikkonen, J., Separation for infinitary logic of uncountable cofinality, to appear.Google Scholar
[16]Oikkonen, J., over , Reports from the Department of Philosophy 3, University of Helsinki, 1983.Google Scholar
[17]Oikkonen, J., How to obtain interpolation for , Logic colloquium '86 (Drake, F. R. and Truss, J. K., editors), North-Holland, 1988, pp. 175–208.Google Scholar
[18]Oikkonen, J., Ehrenfeucht-Fraïssé equivalence of linear ordering, this Journal, vol. 55 (1990), no. 1, pp. 65–73.Google Scholar
[19]Oikkonen, J., Chain models and infinitely deep logics, Reports of the Department of Mathematics, University of Helsinki, 1992, preprint 13.Google Scholar
[20]Svenonius, L., On the denumerable models of theories with extra predicates, The theory of models (Addison, , Henkin, , and Tarski, , editors), North-Holland, 1965, pp. 376–389.Google Scholar
[21]Tuuri, H., Relative separation theorems for , Notre Dame Journal of Formal Logic, vol. 33 (1992), no. 3, pp. 383–401.CrossRefGoogle Scholar
[22]Vaught, R., Descriptive set theory in , Cambridge summer school in mathematical logic, Lecture Notes in Mathematics, vol. 337, Springer-Verlag, 1973, pp. 574–598.CrossRefGoogle Scholar