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

1.1.4 Algebras of boolean signature
Algebras of boolean signature are algebras with signature B = (B, ∨, ¬, ⊥). ∨, ¬, ⊥ are called join,
negation, and bottom respectively. An algebra of boolean signature A is a B-sub-algebra if it is of
the form (X, ∨|, ¬|, ⊥|) where X is a subset of B and ∨|, ¬|, ⊥| are the restrictions of ∨, ¬, ⊥ to
X, moreover X must be such that ∨| ∈ X × X → X and ¬|, ⊥| ∈ X → X. The last requirements
here are to exclude X’s that are not closed under ∨|, ¬|, ⊥|. A boolean homomorphism is
a mapping from an algebra of boolean signature to an algebra of boolean signature that preserves
∨, ¬, ⊥.
The algebras that are of interest in the present paper are each expansions of algebras of boolean
signature. This is to say that they are algebras of boolean signature with extra operations on them.
Those of the algebras that are expansions of boolean algebras turn out to be very hard to axiomatise.
We therefore dispose of with the usual axioms all together. By the representation theorem of Stone
the following is way of defining boolean algebras.
Definition An algebra of boolean signature is a boolean algebra if it is embeddable into
(P(U), ∪, −, ∅), where U is some set,P(U) is the power-set of U, ∪ is union, − is (absolute)
complement and ∅ is the empty set.
A boolean algebra is finitely generated if it is generated by a finite subset of the carrier-set under the
boolean operations. The following property of boolean algebras is central in the present paper.
Lemma 1.1.1 Finitely generated boolean algebras are finite.
Now if U is infinite then a finitely generated sub algebra of (P(U), ∪, −, ∅) is finite but its
elements may be infinite sets. As we seek to make computers work with boolean algebras the
following is useful.
Lemma 1.1.2 Let A = ({⊥, ⊤}, ∨, ¬, ⊥) be the usual two element boolean algebra. Every finite
boolean algebra is isomorphic to an algebra with carrier-set n → {⊥, ⊤} where n ∈ ω and where the
operations are defined component-wise.
To say that the operations on a set of mappings are defined component-wise is to say that
the join of f and g for instance, which for now we denote by (f∨g), is the uniquely determined
operation with the following property. For each i ∈ n it is the case that (f∨g)(i) = f(i) ∨ g(i).
For the particular case, where the mappings are to a two element set, the term bit-wise is sometimes
used.
1.2 Algebras of polyadic signature
Inspired by Tarskis cylindric algebras, Halmos introduced and studied polyadic algebras. Some
papers on these studies are collected in [Hal62]. A particular class of polyadic algebras are polyadic
set algebras of dimension 3, denoted Ps 3 (see definition below). In the present paper these serve
as abstract or generalised models for L 3 sentences. The well known Lindenbaum algebra, L ′ 3/Γ,
of a consistent and complete L ′ 3 theory Γ can be equipped with extra operations related to variable
substitution and existential quantifiers to make it a Ps 3 . It is possible to define interpretation and
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