A generalization of the concept of ω-consistency

Extract

In this paper we consider certain formal properties of deductive systems which, in special cases, reduce to the property of ω-consistency; and we then seek to understand the significance of these properties by relating them to the use of models in providing interpretations of the deductive systems.

The notion of ω-consistency arises in connection with deductive systems of arithmetic. For definiteness, let us suppose that the system is a functional calculus whose domain of individuals is construed as the set of natural numbers, and that the system possesses individual constants ν₀, ν₁, ν₂, … such that ν_i functions as a name for the number i. Such a system is called ω-consistent, if there is no well-formed formula A(x) (in which x is the only free variable) such that A(ν₀), A(ν₁), A(ν₂), … and ∼(x)A(x) are all formal theorems of the system, where A(ν_i) is the formula resulting from A(x) by substituting the constant ν_i for each free occurrence of the individual variable x.

Now consider an arbitrary applied functional calculus F, and let Γ be any non-empty set of its individual constants. In imitation of the definition of ω-consistency, we may say that the system F is Γ-consistent, if it contains no formula A(x) (in which x is the only free variable) such that ⊦ A (α) for every constant α in Γ, and also ⊦ ∼(x)A(x) (where an occurrence of “⊦” indicates that the formula which it precedes is a formal theorem). We easily see that the condition of Γ-consistency is equivalent to the condition that the system F contain no formula B(x) such that ⊦ ∼ B(α) for each α in Γ, and also ⊦ (∃x)B(x).