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Nondiversity in substructures

  • James H. Schmerl (a1)
Abstract

For a model of Peano Arithmetic, let Lt() be the lattice of its elementary substructures, and let Lt+ () be the equivalenced lattice (Lt(),≅), where ≅ is the equivalence relation of isomorphism on Lt(). It is known that Lt+() is always a reasonable equivalenced lattice.

Theorem. Let L be a finite distributive lattice and let (L, E) be reasonable. If 0 is a nonstandard prime model of PA, then 0 has a cofinal extension such that Lt+() ≅ (L,E).

A general method for proving such theorems is developed which, hopefully, will be able to be applied to some nondistributive lattices.

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[1]Abramson, F. G. and Harrington, L. A., Models without indiscernibles, this Journal, vol. 43 (1978), pp. 572600.
[2]Erdős, P. and Rado, R., A combinatorial theorem, Journal of the London Mathematical Society, vol. 25 (1950), pp. 249255.
[3]Erdős, P. and Rado, R., A partition calculus in set theory, Bulletin of the American Mathematical Society, vol. 62 (1956), pp. 427489.
[4]Graham, R. L., Rothschild, B., and Spencer, J. H., Ramsey theorey, John Wiley & Sons, New York, 1990.
[5]Hodges, W., Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.
[6]Hodges, W., A shorter model theory, Cambridge University Press, Cambridge, 1997.
[7]Kossak, R. and Schmerl, J. H., The structure of models of Peano Arithmetic, Oxford Logic Guides, vol. 50, Oxford University Press, Oxford, 2006.
[8]Nešetřil, J. and Rödl, V., The Ramsey property for graphs with forbidden complete subgraphs, Journal of Combinatorial Theory, Series B, vol. 20 (1976), pp. 243249.
[9]Schmerl, J. H., Substructure lattices of models of Peano Arithmetic, Logic colloquium '84 (Manchester, 1984), North-Holland, Amsterdam, 1986, pp. 225243.
[10]Schmerl, J. H., Finite substructure lattices of models of Peano Arithmetic, Proceedings of the American Mathematical Society, vol. 117 (1993), pp. 833838.
[11]Schmerl, J. H., Diversity in substructures, Nonstandard models of arithmetic and set theory, Contemporary Mathematics, vol. 361, American Mathematical Society, Providence, 2004, pp. 145161.
[12]Voigt, B., Canonizing partition theorems: diversification, products, and iterated versions, Journal of Combinatorial Theory, Series A, vol. 40 (1985), pp. 349376.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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