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

We now define a class of automata that can be used to define the atom-structures of certain
Ps n ’s and MsPs n ’s. In this definition we abandon the equivalence relations T and revert to
partitioning by an indexed family {T j } j∈J where J is a finite set. To explain the reason for this
let B n be the boolean algebra generated by our partition. To make an atom-structure of a Ps n
or MsPs n we need to ensure that the image of B n under each cylindrification is a sub-boolean
algebra of B n . The latter appears to require an infinite union or some other higher-order construct,
see Ax4 below.
Definition Let n, p ∈ N where p is a prime power and let J be a finite set. A J-indexed P n -
automaton is a W = (V, π, r, δ, {T j } j∈J ) where
V = (V, . . .) is a vector-space over F p ,
π, r : V → V are called projection and rotation,
δ : V × V → V is called the transition function,
T j ⊆ V for each j ∈ J.
Ax1 If V is finite then (V, π, r) is isomorphic to (M p (n, m), π, r) for some m ∈ N depending
on the size of V.
Ax2 W is a PTPS-automaton using the abstract vector-space, V, as alphabet.
Ax3b W is properly partitioned.
Ax4 For each i < n and j ∈ J there exists a X ⊆ J such that E i (T j ) = ⋃ {T j ′|j ′ ∈ X}.
Again we add a variant with equality, i.e., one that for each i, i ′ < n has a subset of states, D ii ′,
consisting of those states that we run through with some tape whose row i and row i ′ are equal.
Definition Let n, p ∈ N where p is a prime power and let J be a finite set. A J-indexed PE n -
automaton is a W = (W ′ , {D ii ′} i,i ′