Categoricity of theories in Lκω , when κ is a measurable cardinal. Part 1

Saharon Shelah; Oren Kolman

Fundamenta Mathematicae (1996)

  • Volume: 151, Issue: 3, page 209-240
  • ISSN: 0016-2736

Abstract

top
We assume a theory T in the logic L κ ω is categorical in a cardinal λ κ, and κ is a measurable cardinal. We prove that the class of models of T of cardinality < λ (but ≥ |T|+κ) has the amalgamation property; this is a step toward understanding the character of such classes of models.

How to cite

top

Shelah, Saharon, and Kolman, Oren. "Categoricity of theories in Lκω , when κ is a measurable cardinal. Part 1." Fundamenta Mathematicae 151.3 (1996): 209-240. <http://eudml.org/doc/212193>.

@article{Shelah1996,
abstract = {We assume a theory T in the logic $L_\{κω\}$ is categorical in a cardinal λ κ, and κ is a measurable cardinal. We prove that the class of models of T of cardinality < λ (but ≥ |T|+κ) has the amalgamation property; this is a step toward understanding the character of such classes of models.},
author = {Shelah, Saharon, Kolman, Oren},
journal = {Fundamenta Mathematicae},
keywords = {categoricity; classification theory in nonelementary classes; amalgamation property},
language = {eng},
number = {3},
pages = {209-240},
title = {Categoricity of theories in Lκω , when κ is a measurable cardinal. Part 1},
url = {http://eudml.org/doc/212193},
volume = {151},
year = {1996},
}

TY - JOUR
AU - Shelah, Saharon
AU - Kolman, Oren
TI - Categoricity of theories in Lκω , when κ is a measurable cardinal. Part 1
JO - Fundamenta Mathematicae
PY - 1996
VL - 151
IS - 3
SP - 209
EP - 240
AB - We assume a theory T in the logic $L_{κω}$ is categorical in a cardinal λ κ, and κ is a measurable cardinal. We prove that the class of models of T of cardinality < λ (but ≥ |T|+κ) has the amalgamation property; this is a step toward understanding the character of such classes of models.
LA - eng
KW - categoricity; classification theory in nonelementary classes; amalgamation property
UR - http://eudml.org/doc/212193
ER -

References

top
  1. [CK] C. C. Chang and H. J. Keisler, Model Theory, North-Holland, 1973. 
  2. [D] M. Dickmann, Large Infinitary Languages: Model Theory, North-Holland, 1975. Zbl0324.02010
  3. [D1] M. Dickmann, Larger infinitary languages, Chapter IX of Model-Theoretic Logics, J. Barwise and S. Feferman (eds.), Perspect. Math. Logic, Springer, New York, 1985, 317-363. 
  4. [HaSh323] B. Hart and S. Shelah, Categoricity over P for first order T or categoricity for φ L ω 1 ω can stop at k while holding for 0 , . . . , k - 1 , Israel J. Math. 70 (1990), 219-235. 
  5. [HoSh109] W. Hodges and S. Shelah, Infinite games and reduced products, Ann. Math. Logic 20 (1981), 77-108. Zbl0501.03014
  6. [J] T. Jech, Set Theory, Academic Press, 1978. 
  7. [K] H. J. Keisler, Model Theory for Infinitary Logic, North-Holland, 1971. Zbl0222.02064
  8. [L] R. Laver, On Fraïssé's order type conjecture, Ann. of Math. 93 (1971), 89-111. Zbl0208.28905
  9. [MaSh285] M. Makkai and S. Shelah, Categoricity of theories in L κ w , with κ a compact cardinal, Ann. Pure Appl. Logic 47 (1990), 41-97. Zbl0704.03015
  10. [M] M. Morley, Categoricity in power, Trans. Amer. Math. Soc. 114 (1965), 514-518. Zbl0151.01101
  11. [N] M. Nadel, L ω 1 ω and admissible fragments, Chapter VIII of Model-Theoretic Logics, J. Barwise and S. Feferman (eds.), Perspect. Math. Logic, Springer, New York, 1985, 271-316. 
  12. [Re] J. P. Ressayre, Sur les théories du premier ordre catégorique en un cardinal, Trans. Amer. Math. Soc. 142 (1969), 481-505. Zbl0209.30403
  13. [Ro] F. Rowbottom, The Łoś conjecture for uncountable theories, Notices Amer. Math. Soc. 11 (1964), 284. 
  14. [Sh2] S. Shelah, Stable theories, Israel J. Math. 7 (1969), 187-202. Zbl0193.30002
  15. [Sh31] S. Shelah, Solution to Łoś conjecture for uncountable languages, in: Proc. Sympos. Pure Math. 25, Amer. Math. Soc., 1974, 53-74. 
  16. [Sh48] S. Shelah, Categoricity in 1 of sentences in L ω 1 , ω ( Q ) , Israel J. Math. 20 (1975), 127-148. 
  17. [Sh87] S. Shelah, Classification theory for non-elementary classes I: The number of uncountable models of ψ L ω 1 , ω , Parts A, B, Israel J. Math. 46 (1983), 212-240, 241-273. 
  18. [Sh88] S. Shelah, Classification theory for non elementary classes II. Abstract elementary classes, in: Classification Theory, Proc. US-Israel Workshop on Model Theory in Mathematical Logic, Springer, 1987, 419-497. 
  19. [Sh220] S. Shelah, Existence of many L , λ -equivalent, non-isomorphic models of T of power λ, Ann. Pure Appl. Logic 34 (1987), 291-310. 
  20. [Sh300] S. Shelah, Universal classes, in: Classification Theory, Proc. US-Israel Workshop on Model Theory in Mathematical Logic, Springer, 1987, 264-418. 
  21. [Sh420] S. Shelah, Advances in cardinal arithmetic, in: Finite and Infinite Combinatorics in Sets and Logic, N. W. Sauer et al. (eds.), Kluwer Acad. Publ., 1993, 355-383. 
  22. [Sh394] S. Shelah, Categoricity of abstract classes with amalgamation, preprint. 
  23. [Sh472] S. Shelah, Categoricity for infinitary logics II, Fund. Math., submitted. 
  24. [Sh576] S. Shelah, On categoricity of abstract elementary classes: in three cardinals imply existence of a model of the next, preprint. 
  25. [Sh600] S. Shelah, Continuation of [Sh576], in preparation. 
  26. [Sh600] S. Shelah, Classification Theory and the Number of Non-Isomorphic Models, North-Holland, 1978. 
  27. [Sh-a] S. Shelah, Classification Theory and the Number of Non-Isomorphic Models, Classification Theory and the Number of Non-Isomorphic Models, revised, Stud. Logic Found. Math. 92, North-Holland, 1990. 
  28. [Sh-h] S. Shelah, Classification Theory and the Number of Non-Isomorphic Models, Universal classes, preprint. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.