A few remarks on n-infinite forcing companions

We show that the basic properties of Robinson's infinite forcing companions are naturally transmitted to the so called n-infinite forcing com- panions and start with the examination of mutual relations of n-infinite forcing companions of Peano arithmetic. 1. Preliminaries Throughout the article L is a first order language. In general discussions mostly it is irrelevant whether it is with equality or not; however, in some cases, for instance when it comes to finite models, the supposition of the existence of the equality relation could be of significance - see 2.6. For a theory T of the language L, µ(T ) will be the slass of all its models (as usual, by a theory we assume a consistent deductively closed set of sentences - thus, T ϕ means ϕ ∈ T ). By Σn-formula we mean any formula equivalent to a formula in prenex normal form whose prenex consists of n blocks of quantifiers, the first one is the block of existential quantifiers (Πn-formulas are defined analoguosly). The models (of the language L) will be denote by A,B... , while their domains will be A, B, ... . For a model A, Diagn(A) is the set of all Σn-, Πn-senteneces of the language L(A) (the simple expansion of the language L obtained by adding a new set of constants which is in one to one correspendence with domain A )w hich hold in A. In particular, for n = 0, Diag0(A) is not the diagram of A in the sense in which it is used in model theory, but this difference is of no importance for the text (the same situation we had when we were dealing with the generalization of finite forcing). As usual, we will not distinguish an element a from A and to it the corresponding constant. If A is a submodel of B and (B ,a )a∈A Diagn(A), we say that A is an n-elementary submodel of B (i.e., that B is an n-elementary extension of A), in notation A ≺n B. In general, A is n-embedded in B if for some embedding f of A into B, f (A )i s ann-elementary submodel of B.A Σ n+1-chain of models is a chain of models A0 < A1 < ··· < Aα < ··· , α<γ , where for each