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HENKIN CONSTRUCTIONS OF MODELS WITH SIZE CONTINUUM

  • JOHN T. BALDWIN (a1) and MICHAEL C. LASKOWSKI (a2)

Abstract

We describe techniques for constructing models of size continuum in ω steps by simultaneously building a perfect set of enmeshed countable Henkin sets. Such models have perfect, asymptotically similar subsets. We survey applications involving Borel models, atomic models, two-cardinal transfers and models respecting various closure relations.

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[1]Ackerman, N., Freer, C., and Patel, R., Invariant measures concentrated on countable structures. Forum of Mathematics Sigma, vol. 4 (2016), no. e17, p. 59.
[2]Baldwin, J. T., Categoricity, University Lecture Notes, vol. 51, American Mathematical Society, Providence, USA, 2009.
[3]Baldwin, J. T., The explanatory power of a new proof: Henkin’s completeness proof, Philosophy of Mathematics: Truth, Existence and Explanation (Piazza, M. and Pulcini, G., editors), FilMat 2016 Studies in the Philosophy of Mathematics, Springer-Verlag, Berlin, 2018, pp. 147162.
[4]Bays, M., Categoricity results for exponential maps of 1-dimensional algebraic groups & Schanuel Conjectures for Powers and the CIT, Ph.D. thesis, Oxford, 2009. Available at http://people.maths.ox.ac.uk/∼bays/dist/thesis/.
[5]Baldwin, J. T., Laskowski, M. C., and Shelah, S., Constructing many atomic models in${\aleph _1}$.. Journal of Symbolic Logic, vol. 81 (2016), pp. 11421162.
[6]Bays, M. and Zilber, B. I., Covers of multiplicative groups of an algebraically closed field of arbitrary characteristic. Bulletin of the London Mathematical Society, vol. 43 (2011), pp. 689702.
[7]Henkin, L., The completeness of the first-order functional calculus. Journal of Symbolic Logic, vol. 14 (1949), pp. 159166.
[8]Hjorth, G., Knight’s model, its automorphism group, and characterizing the uncountable cardinals. Journal of Mathematical Logic, vol. 8 (2002), pp. 113144.
[9]Hjorth, G., A note on counterexamples to Vaught’s conjecture. Notre Dame Journal of Formal Logic, vol. 48 (2007), no. 1, pp. 4951.
[10]Hafner, J. and Mancosu, P., The varieties of mathematical explanation, Visualization, Explanation, and Reasoning Styles in Mathematics (Mancosu, P., Jorgensen, K. F., and Pedersen, S., editors), Springer, Berlin, 2005, pp. 251–249.
[11]Hrushovski, E. and Shelah, S., Stability and omitting types. Israel Journal of Mathematics, vol. 74 (1991), pp. 289321.
[12]Hirschfeldt, D., Shore, R., and Slaman, T., The atomic model theorem and type omitting. Transactions of the American Mathematical Society, vol. 361 (2009), pp. 58055837.
[13]Keisler, H. J., Model Theory for Infinitary Logic, North-Holland, Amsterdam, 1971.
[14]Kirby, J., On quasiminimal excellent classes. Journal of Symbolic Logic, vol. 75 (2010), pp. 551564.
[15]Kim, B., Kim, H.-J., and Scow, L., Tree indiscernibilities, revisited. Archive for Mathematical Logic, vol. 53 (2015), no. 2–14, pp. 211232.
[16]Knight, J. F., A complete ${L_{{\omega _1},\omega }}$-sentence characterizing${\aleph _1}$.. Journal of Symbolic Logic, vol. 42 (1977), pp. 151161.
[17]Kueker, D. W., Uniform theorems in infinitary logic, Logic Colloquium 77 (Macintyre, A., Pacholski, L., and Paris, J., editors), North Holland, Amsterdam, 1978, pp. 161170.
[18]Lachlan, A. H., A property of stable theories. Fundamenta Mathematicae, vol. 77 (1972), pp. 920.
[19]Laskowski, M. C. and Shelah, S., On the existence of atomic models. Journal of Symbolic Logic, vol. 58 (1993), pp. 11891194.
[20]Montalbán, A. and Nies, A., Borel structures, a brief survey, Effective Mathematics of the Uncountable (Greenberg, N., Hamkins, J. D., Hirschfeldt, D., and Miller, R., editors), Lecture Notes in Logic, vol. 41, Association of Symbolic Logic/Cambridge University Press, 2013, pp. 124134.
[21]Resnik, M. and Kushner, D., Explanation, independence, and realism in mathematics. The British Journal for the Philosophy of Science, vol. 38 (1987), pp. 141158.
[22]Rucker, R., White Light, Ace, New York, 1980.
[23]Shelah, S., Categoricity in ${\aleph _1}$of sentences in.${L_{{\omega _1},\omega }}\left( Q \right)$. Israel Journal of Mathematics , vol. 20 (1975), pp. 127148. Sh index 48.
[24]Shelah, S., A two-cardinal theorem. Proceedings of the American Mathematical Society, vol. 48 (1975), pp. 207213. Sh index 37.
[25]Shelah, S., A two-cardinal theorem and a combinatorial theorem. Proceedings of the American Mathematical Society, vol. 62 (1976), pp. 134136. Sh index 49.
[26]Shelah, S., Classification Theory and the Number of Nonisomorphic Models, North-Holland, Amsterdam, 1978.
[27]Shelah, S., Classification theory for nonelementary classes. I. The number of uncountable models of $\psi \in {L_{{\omega _1}\omega }}$part A . Israel Journal of Mathematics , vol. 46 (1983), no. 3, pp. 212240. Sh index 87a.
[28]Shelah, S., Classification theory for nonelementary classes. II. The number of uncountable models of $\psi \in {L_{{\omega _1}\omega }}$part B . Israel Journal of Mathematics , vol. 46 (1983), no. 3, pp. 241271. Sh index 87b.
[29]Shelah, S., Borel sets with large squares. Fundamenta Mathematica, vol. 159 (1999), pp. 150. Sh index 522.
[30]Shelah, S. and Väänänen, J., Recursive logic frames. Mathematical Logic Quarterly, vol. 52 (2006), pp. 151164.
[31]Vaught, R. L., Denumerable models of complete theories, Infinitistic Methods, Proceedings of the Symposium on the Foundations of Mathematics, Warsaw, 1959, Państwowe Wydawnictwo Naukowe, Warsaw, 1961, pp. 303321.
[32]Zilber, B. I., A categoricity theorem for quasiminimal excellent classes, Logic and its Applications (Blass, A. and Zhang, Y., editors), Contemporary Mathematics, vol. 380, American Mathematical Society, Providence, RI, 2005, pp. 297306.

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