Two theories with axioms built by means of pleonasms

Extract

This paper contains examples T₁ and T₂ of theories which answer the following questions:

(1) Does there exist an essentially undecidable theory with a finite number of non-logical constants which contains a decidable, finitely axiomatizable subtheory?

(2) Does there exist an undecidable theory categorical in an infinite power which has a recursive set of axioms? (Cf. [2] and [3].)

The theory T₁ represents a modification of a theory described by Myhill [7]. The common feature of theories T₁ and T₂ is that in both of them pleonasms are essential in the construction of the axioms.

Let T₁ be a theory with identity = which contains one binary predicate R(x, y) and is based on the axioms A₁, A₂, A₃, B₁, B₂, B₃, B₄, C_nm which follow.

A₁: x = x. A₂: x = y ⊃ y = x. A₃: x = y ∧ y = z ⊃ x = z.

(Axioms of identity.)

B₁: R(x, x). B₂: R(x, y) ⊃ R(y, x). B₃: R (x, y) ∧ R(y, z) ⊃ R(x, z).

(Axioms of equivalence.)

B₄: x = y ⊃ [R(z, x) ≡ R(z,y)].

Let φ_n be the formula

which express that there is an abstraction class of the relation R which has exactly n elements.

Let f(n) and g(n) be two recursive functions which enumerate two recursively inseparable sets [5], and call these sets X₁ and X₂.

We now specify the axioms C_mm.

It is obvious that the set composed of the formulas A₁−A₃, B₁−B₄, C_nm (n,m = 1,2, …) is recursive.

The theory T₁ is essentially undecidable; for if there were a complete and decidable extension T′₁ (of it, then the recursive sets Z = {n: φ_n is provable in T′₁} and Z′ = {n: ∼φ_n is provable in T′₁} would separate the sets X₁ and X₂.