Model-theoretic properties characterizing peano arithmetic

Journal of Symbolic Logic 56 (3):949-963 (1991)
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Abstract

Let $\mathscr{L} = \{0, 1, +, \cdot, <\}$ be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order L-theory containing IΔ0 + exp such that every complete extension T of it has a countable model K satisfying. (i) K has no proper elementary substructures, and (ii) whenever $L \prec K$ is a countable elementary extension there is $\bar{L} \prec L$ and $\bar{K} \subseteq_\mathrm{e} \bar{L}$ such that $K \prec_{\mathrm{cf}}\bar{K}$ . Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic

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The Theory of $\kappa$ -like Models of Arithmetic.Richard Kaye - 1995 - Notre Dame Journal of Formal Logic 36 (4):547-559.

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