We follow the notation of [6, §4.3]. We start by defining our languages. We consider certain infinitary languages that are extensions of first order languages. A language Λ is a triple (α, R, ), where α is a countable infinite ordinal. and R are functions with common domain β, β a cardinal. That is, dom(R) = dom( ) = β. α specifies the number of available variables and R = (R i ) i<β is the sequence of relation symbols with domain β. is a function from β to α. For i < β, (i) is the rank of R i . The cardinality of this language is defined to be β. All languages considered in this paper are countable, i. e. β ≤ ω. We do not allow function symbols. When (i) is finite for all i, then we are dealing with an ordinary first order language. But we shall deal with more general languages, namely languages where α \ (i) is infinite. Such languages are called variable rich in [11]. In particular, for such languages the arity of atomic formulas may be infinite. However, formulas are built up from atomic formulas the usual way, using =, ¬, ∧, ∨, and ∃x. In other words, only quantification on finitely many variables is allowed. Since only finite conjunctions and disjunctions are allowed, only finitely many relation symbols can occur in formulas. But let us specify our atomic formulas. An atomic formula is either an equation or one of the formwhere η < β. Such atomic formulas are called restricted atomic formulas, meaning that the variables occur only in their natural order. We allow only these. A restricted formula, or a formula for short, is one whose atomic subformulas are restricted (for first order logic this is not a restriction because any (ordinary) formula is equivalent to a restricted one). We let Fm Λ r stand for the set of all Λ (restricted) formulas. We use the proof system r,Λ for our language Λ introduced in [6, §4.3]. r,Λ is complete for first order languages. Let Ax ⊆ Fm Λ r and ϕ ∈ Fm Λ r . We write Ax r,Λ ϕ or even simply Ax ϕ if ϕ can be derived from Ax by the above mentioned proof system. A theory T is a set of formulas. T is consistent if not T ⊥.After having specified our syntax, we now turn to semantical notions. Our models are not the standard ones. In more detail: Definition 1.1(i) Let α be an ordinal and M be a set. Then α M denotes the set of all functions from α to M . A weak space of dimension α and base M is a set of the form {s ∈ α M : |{i ∈ α : s i = p i }| < ω} for some p ∈ α M . We denote this set by α M (p) .
