Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Lectures on Infinitary Model Theory
  • Cited by 6
Publisher:
Cambridge University Press
Online publication date:
August 2016
Print publication year:
2016
Online ISBN:
9781316855560
DOI:
https://doi.org/10.1017/CBO9781316855560
Subjects:
Mathematics, Logic, Categories and Sets, Algebra, Programming Languages and Applied Logic
Series:
Lecture Notes in Logic (46)
Information

Book description

Infinitary logic, the logic of languages with infinitely long conjunctions, plays an important role in model theory, recursion theory and descriptive set theory. This book is the first modern introduction to the subject in forty years, and will bring students and researchers in all areas of mathematical logic up to the threshold of modern research. The classical topics of back-and-forth systems, model existence techniques, indiscernibles and end extensions are covered before more modern topics are surveyed. Zilber's categoricity theorem for quasiminimal excellent classes is proved and an application is given to covers of multiplicative groups. Infinitary methods are also used to study uncountable models of counterexamples to Vaught's conjecture, and effective aspects of infinitary model theory are reviewed, including an introduction to Montalbán's recent work on spectra of Vaught counterexamples. Self-contained introductions to effective descriptive set theory and hyperarithmetic theory are provided, as is an appendix on admissible model theory.

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
REFERENCES
[1] J., Baldwin, The Vaught Conjecture: Do uncountable models count?, Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 79–92.
[2] J., Baldwin, Categoricity, AMS University Lecture Series, AMS, 2009.
[3] J., Baldwin, Amalgamation, absoluteness, and categoricity, Proceedings of the 11th Asian Logic Conference, World Scientific Publishing, Hackensack, NJ, 2012, pp. 22–50.
[4] J., Baldwin, S., Friedman, M., Koerwien, and M., Laskowski, Three red herrings around Vaught's Conjecture, Transactions of the American Mathematical Society, vol. 268 (2016), pp. 3673–3694.
[5] J., Baldwin and P., Larson, Iterated elementary embeddings and the model theory of infinitary languages, Annals of Pure and Applied Logic, vol. 167 (2016), pp. 309–334.
[6] J., Barwise, The Syntax and Semantics of Infinitary Languages, Lecture Notes in Mathematics, vol. 72, Springer-Verlag, Berlin and New York, 1968, pp. 166–181.
[7] J., Barwise, Back and forth through infinitary logics, Studies in Model Theory (M., Morley, editor), MAA, 1973.
[8] J., Barwise, Admissible Sets and Structures, Springer-Verlag, Berlin, 1975.
[9] J., Barwise and S., Feferman, Handbook of Model Theoretic Logics, Springer-Verlag, Berlin, 1985.
[10] J., Baumgartner, The Hanf number for complete sentences (without GCH), The Journal of Symbolic Logic, vol. 39 (1974), pp. 575–578.
[11] M., Bays, M., Gavrilovich, and M., Hils, Some definability results in abstract Kummer theory, International Mathematics Research Notices, vol. 14 (2014), pp. 3975–4000.
[12] M., Bays, B., Hart, T., Hyttinen, M., Kesälä, and J., Kirby, Quasiminimal structures and excellence, Bulletin of the London Mathematical Society, vol. 46 (2014), pp. 155–163.
[13] M., Bays, B., Hart, and A., Pillay, Universal covers of commutative finite Morley rank groups, preprint.
[14] M., Bays and J., Kirby, Excellence and uncountable categoricity of Zilber's exponential fields, preprint. 175
[15] M., Bays and B., Zilber, Covers of multiplicative groups of algebraically closed fields of arbitrary characteristic, Bulletin of the London Mathematical Society, vol. 43 (2011), pp. 689–702.
[16] J., Burgess, Equivalences generated by families of Borel sets, Proceedings of the American Mathematical Society, vol. 69 (1978), pp. 323–326.
[17] W., Calvert, J., Knight, and J., Millar, Computable trees of Scott rank and computable approximation, The Journal of Symbolic Logic, vol. 71 (2006), pp. 283–298.
[18] M., Dickmann, Larger infinitary languages, in [9].
[19] H., Friedman and R., Jensen, A note on admissible ordinals, in [6].
[20] S., Friedman, T., Hyttinen, and M., Koerwien, The nonabsoluteness of model existence in uncountable cardinals for, Notre Dame Journal of Formal Logic, vol. 54 (2013), pp. 137–151.
[21] R., Gandy, G., Kreisel, and W. W., Tait, Set existence, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 8 (1960), pp. 577–582.
[22] S., Gao, On automorphism groups of countable structures, The Journal of Symbolic Logic, vol. 63 (1998), pp. 891–896.
[23] S., Gao, Invariant Descriptive Set Theory, Pure and Applied Mathematics, vol. 293, CRC Press, Boca Raton, 2009.
[24] V., Harnik and M., Makkai, A tree argument in infinitary model theory, Proceedings of the American Mathematical Society, vol. 67, 1977, pp. 309–314.
[25] L., Harrington, Analytic determinacy and, The Journal of Symbolic Logic, vol. 43 (1978), pp. 685–693.
[26] M., Hindry and J., Silverman, Diophantine Geometry. An Introduction, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000.
[27] P., Hinman, Recursion Theoretic Hierarchies, Springer-Verlag, Berlin, New York, 1978.
[28] G., Hjorth, Knight's model, its automorphism group, and characterizing the uncountable cardinals, Journal of Mathematical Logic, vol. 2 (2002), pp. 113– 144.
[29] G., Hjorth, A note on counterexamples to the Vaught conjecture, Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 49–51.
[30] T., Jech, Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[31] A., Kanamori, The Higher Infinite, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994.
[32] I., Kaplinsky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor, 1954.
[33] M., Kaufmann, The quantifier “there exist uncountably many” and some of its relatives, in [9].
[34] A. S., Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995.
[35] A. S., Kechris, D. A., Martin, and R., Solovay, Introduction to Q-theory, Cabal Seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 199–282.
[36] H. J., Keisler, Logic with the quantifier “there exist uncountably many”, Annals of Mathematical Logic, vol. 1 (1970), pp. 1–93.
[37] H. J., Keisler, Model Theory of Infinitary Languages, North-Holland, Amsterdam, 1971.
[38] H. J., Keisler and M., Morley, Elementary extensions of models of set theory, Israel Journal of Mathematics, vol. 6 (1968), pp. 49–65.
[39] J., Kirby, On quasiminimal excellent classes, The Journal of Symbolic Logic, vol. 75 (2010), pp. 551–564.
[40] J., Knight, A complete-sentence characterizing, The Journal of Symbolic Logic, vol. 42 (1977), pp. 59–62.
[41] J., Knight, Prime and atomic models, The Journal of Symbolic Logic, vol. 43 (1978), pp. 385–393.
[42] J., Knight, A., Montalbán, and N., Schweber, Computable structures in generic extensions, preprint.
[43] D., Kueker, Back-and-forth arguments in infinitary languages, Infinitary Logic: In Memoriam Carol Karp (D., Kueker, editor), Lecture Notes in Mathematics, vol. 72, Springer-Verlag, Berlin, 1975.
[44] D., Kueker, Countable approximations and Löwenheim–Skolem theorems, Annals of Mathematical Logic, vol. 11 (1977), pp. 57–103.
[45] D., Kueker, Uniform theorems in infinitary logic, Logic Colloquium ‘77, Studies in Logic and the Foundations of Mathematics, vol. 96, North-Holland, Amsterdam, New York, 1978.
[46] K., Kunen, SetTheory: AnIntroduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1983.
[47] P., Larson, Axiomatizing Scott processes, preprint.
[48] M. C., Laskowski and S., Shelah, On the existence of atomic models, The Journal of Symbolic Logic, vol. 58 (1993), pp. 1189–1194.
[49] M., Makkai, An “admissible” generalization of a theorem on countable sets of reals with applications, Annals of Mathematical Logic, vol. 11 (1977), pp. 1–30.
[50] M., Makkai, Infinitary Logic, Handbook of Mathematical Logic (J., Barwise, editor), Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland, Amsterdam, New York, Oxford, 1977.
[51] M., Makkai, An example concerning Scott heights, The Journal of Symbolic Logic, vol. 46 (1981), pp. 301–318.
[52] J., Malitz, The Hanf number for complete-sentences, in [6].
[53] R., Mansfield, Omitting types: application to descriptive set theory, Proceedings of the American Mathematical Society, vol. 47 (1975), pp. 198– 200.
[54] R., Mansfield and G., Weitkamp, Recursive Aspects of Descriptive Set Theory, Oxford University Press, Oxford, 1985.
[55] D., Marker, Basic definability notions for, appendix to [3].
[56] D., Marker, Model Theory: An Introduction, Springer Graduate Texts in Mathematics, Springer, Berlin, 2002.
[57] D. A., Martin, The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687–689.
[58] D. A., Martin and A. S., Kechris, Infinite games and effective descriptive set theory, Analytic Sets (C. A., Rogers et al., editors), Academic Press, London, 1980.
[59] A., Montalbán, A computability theoretic equivalent to Vaught's conjecture, Advances in Mathematics, vol. 235 (2013), pp. 56–73.
[60] M., Morley, The number of countable models, The Journal of Symbolic Logic, vol. 35 (1970), pp. 14–18.
[61] Y., Moschovakis, Descriptive Set Theory, North Holland, Amsterdam, 1980.
[62] M., Nadel, and admissible fragments, in [9].
[63] M., Nadel, More Löwenheim–Skolem results for admissible sets, Israel Journal of Mathematics, vol. 18 (1974), pp. 53–64.
[64] M., Nadel, Scott sentences and admissible sets, Annals of Mathematical Logic, vol. 7 (1974), pp. 267–294.
[65] J.-P., Ressarye, Models with compactness properties relative to an admissible language, Annals of Mathematical Logic, vol. 11 (1977), pp. 31–55.
[66] G., Sacks, Metarecursively enumerable sets and admissible ordinals, Bulletin of the American Mathematical Society, vol. 72 (1966), pp. 59–64.
[67] G., Sacks, On the number of countable models, Southeast Asian Conference on Logic (Singapore, 1981), Studies in Logic and the Foundations of Mathematics, vol. 111, North-Holland, Amsterdam, 1983, pp. 185–195.
[68] G., Sacks, Higher Recursion Theory, Springer-Verlag, Berlin, 1990.
[69] S., Shelah, Categoricity in of sentences in, Israel Journal of Mathematics, vol. 20 (1975), pp. 127–148.
[70] S., Shelah, Classification theory and the number of nonisomorphic models, vol. 92 (1978).
[71] S., Shelah, The number of pairwise non-elementarily-embeddable models, The Journal of Symbolic Logic, vol. 54 (1989), pp. 1431–1455.
[72] S., Shelah, Borel sets with large squares, Fundamenta Mathematicae, vol. 159 (1999), pp. 1–50.
[73] S., Shelah, L., Harrington, and M., Makkai, A proof of Vaught's conjecture for-stable theories, Israel Journal of Mathematics, vol. 49 (1984), pp. 259–280.
[74] J., Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Annals of Mathematical Logic, vol. 18 (1980), pp. 1–28.
[75] T., Slaman and J., Steel, Definable functions on degrees, Cabal Seminar 81–85, Lecture Notes in Mathematics, vol. 1333, Springer, Berlin, 1988, pp. 37–55.
[76] I., Souldatos, Characterizing the powerset by a complete (Scott) sentence, Fundamenta Mathematicae, vol. 222 (2013), pp. 131–154.
[77] I., Souldatos, Notes on cardinals that are characterizable by a complete (Scott) sentence, Notre Dame Journal of Formal Logic, vol. 55 (2014), pp. 533–551.
[78] J., Steel, On Vaught's conjecture, Cabal Seminar 76–77, Lecture Notes in Mathematics, vol. 689, Springer, Berlin, 1978, pp. 193–208.
[79] B., Zilber, Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic, vol. 132 (2005), pp. 67–95.
[80] B., Zilber, Covers of the multiplicative group of an algebraically closed field of characteristic zero, Journal of the London Mathematical Society, vol. 74 (2006), pp. 41–58.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.