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Observations concerning elementary extensions of ω-models. II

This paper contains some observations on models for A2-second-order arithmetics. It arose from unsuccessful attempts to prove the Conjecture 1 (see at the end of the paper). Two partial solutions are given in this paper. We also give a sketch of a new proof of a classical result of descriptive set-theory referred to in [3]. Unless otherwise specified the word “model” means “ω-model for second-order arithmetics with the axiom scheme of choice,” i.e., “ω-model of A2” in the terminology of [3]. We say that a model M1 is shorter than M2 iff there is in M2 a relation A which is a well-ordering in M2 such that every relation B which is a well-ordering in M1 is similar to an initial segment of A. We denote by M0 the principal model of A2, i.e., a model which has only standard integers and contains all sets of integers.