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Nondiversity in substructures

  • James H. Schmerl (a1)

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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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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