Constructing Sentences

iii.    Include in D a sentence to say that each integer is the successor of just one integer.

        [Succ]    "z$x zºx' Ù "xyz[zºx' Ù zºy' ® xºy]

iv.    Include in D a sentence to say that if p, q are Naturals, p ¹ q, then D |^_ "x x^p ¹ x^q.

        [Diff]    "xyz[x<y Ù y<z ® x<z] Ù "xy[x'ºy ® x<y] Ù "xy[x<y ® xºy]

4. Construct the sentence H that says (under A) that TM will halt.

    .    We know that TM will halt if at t it is in q_i scanning S_j

         and its machine table has no quadruple beginning (q_i, S_j, ...)

    .    Such a missing instruction gives rise to a sentence including Q_itx Ù S_jtx

    .    To state that TM gets to some such position we create the disjunction of all sentences

         $tx[Q_itx Ù S_jtx] for those missing instructions.

    .    If there are no missing instructions thatn TM will not halt

         and for H we can use any falsehood, such as e ¹ e.