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Extra resources for Computability: Turing, Goedel, Church, and beyond

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R. 1961. Minds, machines and Gödel. Philosophy 36:112–127. Lucas, J. R. 1996. Minds, machines and Gödel: A retrospect. In Machines and Thought: The Legacy of Alan Turing, Vol. 1, ed. P. Millican and A. Clark, 103–124. Oxford: Clarendon Press. McCulloch, W. , and W. Pitts. 1943. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics 5:115–133. Minsky, M. L. 1967. Computation: Finite and Infinite Machines. Englewood Cliffs, NJ: Prentice-Hall. Penrose, R.

Yuri’s proof is quite short, but it is a wonderful tapestry, delicate and beautiful. Putting the pieces together, we now had: DPRM Theorem2 If S ⊆ Nk is listable, then there is a polynomial with integer coefficients p(t1, t2, ... , tk, x1, x2, ... , xn) such that < t1 , t2 , … , t k > ∈ S ⇔ (∃x1 , … , xn )[ p(t1 , t2 , … , t k , x1 , x2 , … , xn ) = 0]. Because there are listable sets that are not computable, it follows that there is a polynomial p such that {r ∈ N | (∃x1 , x2 , … , xn ) p(r , x1 , x2 , … , xn ) = 0} is not computable.

For i = 0, 1, ... , m, using the substitutivity of equality, we obtain A ( " x )[( x = i* ) ⊃ [α (i* ) ⊃ α ]]. The result follows. 10 Let α contain the single free variable x, and suppose that A ¬α (i* ), for i = 0, 1, ... , m. Then, A ¬( $x )[( x ≤ m* ) ∧ α ].

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