1.    Construct a language L appropriate to our proposed D and H.

        For each state q_i let there be a 2-place predicate symbol Q_i,

        For each symbol S_j let there be a 2-place predicate symbol S_j,

        Include logical symbols,

        Include equality predicate symbol º,

        Include 2-place predicate symbol <,

        Include 1-place function symbol ',

        Include constant symbol e.

2.    There is an intended interpretation A of the language L for the sentences in D and H.

        .    | A | = Integers

        .    e^A = 0 

        .    '^A is the successor function

        .    Val (Q_itx, A, a) = T iff at time t TM is in state q_i scanning cell x

        .    Val (S_jtx, A, a) = T iff at time t symbol S_j is in cell x

        .    Val (x<y, A, a) = T iff x < y

3.    Construct the sentences of D.

        Note that our intended interpretation distinguishes between numbers that are to stand for time indexes and numbers that are to stand for cell indexes. We shall continue s, t and x, y, z respectively to indicate this distinction.

        We shall also suppose that the only symbols that the TM reads or writes are S₀, ..., S_r.