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The theory of \(\kappa\)-like models of arithmetic. (English) Zbl 0848.03019
A linearly ordered model is \(\kappa\)-like if it is of power \(\kappa\), and each of its proper initial segments is of cardinality smaller than \(\kappa\). In the search for the theory of all \(\kappa\)-like models of PA, the author describes axiom schemes true in all \(\kappa\)-like models of PA, and discusses their model-theoretic properties. The schemes are suitable formalizations of: the statement that in a \(\kappa\)-like model there is no one-to-one map from the model to a proper initial segment; various forms of the pigeonhole principle; collection scheme; an indiscernibility schema related to a general theorem of Keisler on \(\kappa\)-like models for singular strong limits \(\kappa\). The concept of tree indiscernibility related to the work of Paris and Mills on cardinalities of initial segments of models of PA is also discussed. The paper is intended to be read in conjunction with another paper of the author’s [“Constructing \(\kappa\)-like models of arithmetic”, J. Lond. Math. Soc. (to appear)].

MSC:
03C62 Models of arithmetic and set theory
03F30 First-order arithmetic and fragments
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