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

Proof: We turn a fragment of first-order language into an algebra L crc
n of dMsP n -signature as
follows. The sort with index n consist of boolean combinations of atomic formulae built from
a countably infinite supply of n-ary relation symbols. The sort with index i consist of boolean
combinations of formulae of the form ∃x i φ, where φ is of the sort with index i + 1. See see
A. Rognes [Rog09] for further details. Interpretations of L crc
n -formulae are nothing more than
dMsP n -homomorphisms from L crc
n to other algebras of dMsP n -signature. An interpretation f
is satisfying for a sentence φ if f 0 (φ) = ⊤.
(⇒): Assume that A = (B n , . . .) has purely infinite spectrum. We construct an infinity axiom
φ of L crc
n , that has a satisfying interpretation f in A, by describing A up to isomorphism. For
this construction we need one n-ary relation symbol, R, for each element of B n . The satisfying
interpretation of φ is uniquely determined by mapping the atomic formula R a (x 0 , . . . , x n−1 ) to
the element a ∈ B n . Since A is simple there is for each element a ∈ B i a formula ψ such that
f i (ψ) = a. This allows us to describe, in L crc
n , what the result of any operation of A is, on any
arguments. Since A is finite, the conjunction of these descriptions is again a sentence in L crc
n . The
conjunction is clearly satisfied by A and if it had a model over a finite set U then A would be
embeddable into the full dMsPs n over U. But A was assumed to have purely infinite spectrum
so the conjunction is an infinity axiom.
(⇐): Assume now that φ is an infinity axiom of L crc
n and let f be a satisfying interpretation, i.e.,
a homomorphism from L crc
n to A with the property that f 0 (φ) = ⊤. Assume, for the purpose of
arriving at a contradiction, that there is a finite cardinal number in spec(A). This means that there
is a finite set U and a homomorphism f ′ from A into the full dMsPs n over U. The composition
of f with f ′ now is an interpretation of φ in the full dMsPs n over U. The composition maps φ
to ⊤ since homomorphisms preserve ⊤. But then φ has a model whose carrier set is the finite set
U. This contradicts the assumption that φ is an infinity axiom.
qed
3.2 The polyadic atom-structure and the h-complex algebra
We define the polyadic analog of atom-structure, following the terminology of R. Hirsch and I. M.
Hodkinson, [Hod97] [HH09], who work with relation and cylindric algebras rather than polyadic
algebras. The polyadic version of atom-structures are used in the definition of complex algebra in
the book of L.Henkin, J.D.Monk and A.Tarski [MHT85], but called relational structure. Finite
polyadic atom-structures correspond to (finite many-sorted) polyadic algebras but atom-structures
are preferable as objects of computation as an atom-structure with k elements has a corresponding
algebra with 2 k elements. Since n is fixed throughout, we simply write atom-structure to mean
n-dimensional polyadic atom-structure in this section.
3.2.1 Atom-structures and n-homomorphisms
Here the atom-structure is formally defined. Moreover we define a special kind of homomorphism
designed to correspond to dMsP n -embeddings. The homomorphisms are called n-homomorphisms
and are believed to be new.
Definition Let n ∈ N. An n-dimensional polyadic atom-structure is a tuple
(H, ⊳ r , ⊳ p , ⊳ s , E 0 , . . . , E n−1 ), where
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