On the methods of mechanical non-theorems (latest version)

1.1 Introduction
In this setting a disprover is a procedure that terminates on input of satisfiable first-order sentences
only. The purpose being to use the disprover in conjunction with a procedure that terminates on
input of inconsistent first-order sentences only, in attempts at deciding whether given sentences are
satisfiable by means of a computer.
The present paper describes a method for devising disprovers, some of which have been implemented.
Focus is on why disprovers deviced according to the method work in principle and to what
extent. We rely on established results on the Entscheidungsproblem and use techniques common
in algebraic logic for proofs. A particular class of many-sorted polyadic set algebras suitable for the
task is introduced.
One needs a decision procedure for some first-order theory to devise a disprover according to the
method. By means of the decision procedure a finite many-sorted polyadic set algebra is computed.
Such an algebra forms the basis of one disprover, which on input of a first-order sentence works
by exhaustive search for satisfying interpretations for the sentence in the algebra and in successively
refined versions of the algebra.
Depending on the decision procedure, resulting disprovers can be made to recognise satisfiable
sentences that are not finitely satisfiable. Such sentences are called infinity axioms, and here is an
example.
∀x∃yRxy∧ ∀x¬Rxx∧ ∀x∀y∀zRxy ∧ Ryz → Rxz
The set of sentences recognised by each disprover, turns out to be non-recursive. This is a
consequence of a result of Büchi, sharpening Trakhtenbrots theorem, together with the ability to
recognise any finitely satisfiable sentence of a substantial fragment of first-order language [Büc62].
This implies that no decidable class of sentences covers the set of sentences recognised by any one
of the disprovers described here. Since finite unions of decidable sentence classes are decidable, no
finite union of decidable sentence classes will cover any of the recognised sets either.
As any disprover can be repaired in a most ad hoc way so as to recognise any given infinity axiom,
the following naturalness property is shown. The set of sentences recognised by each disprover is
closed under logical equivalence, within the aforementioned fragment. This property is not shared
with procedures that work by first checking whether the input is syntactically equal to one of a
finite set of satisfiable sentences and then go on with, say, search for satisfying interpretations over
finite sets.
Procedures that do search for satisfying interpretations over finite sets are said to do finite model
search. Each of the disprovers described here externally share the ability to recognise any finitely
satisfiable sentence of a substantial fragment of first-order language and the naturalness property
with finite model search procedures. Also internally there is some resemblance. We therefore use
the term generalised finite model search, for describing how the presented disprovers work. Those
of the disprovers that recognise infinity axioms represent a strict generalisation of finite model
search.
The model search generalisation and the results about it, work for conjunctions of purely relational
prenex sentences whose length of quantifier prefix is limited by a constant. For the following
reasons that constant is fixed at 3 throughout. It has made implementation feasible. To some extent
it makes geometric inspection feasible. It economises notation. There is no loss of generality, in
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