Computability: Turing, Goedel, Church, and beyond by Copeland B.J., et al. (eds.)

By Copeland B.J., et al. (eds.)

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Super Turing-machines. Complexity 4:30–32. Copeland, B. J. 2000. Narrow versus wide mechanism. Journal of Philosophy 97:5–32. Reprinted in Computationalism: New Directions, ed. M. Scheutz, 59–86. Cambridge: MIT Press, 2002. Copeland, B. J. 2002a. Accelerating Turing machines. Minds and Machines 12:281–301. Copeland, B. J. 2002b. Hypercomputation. In Hypercomputation, special issue, Minds and Machines 12:461–502. Copeland, B. , ed. 2002–2003. Hypercomputation. Special issue, Minds and Machines 12, 13.

Epistemic two-dimensional semantics. Philosophical Studies 18:153–226. Church, A. 1936. An unsolvable problem of elementary number theory. American Journal of Mathematics 58:345–63. Reprinted in Davis, The Undecidable, 88–107. Church, A. 1937a. Review of Turing (1936). Journal of Symbolic Logic 2:42–43. Church, A. 1937b. Review of Post (1936). Journal of Symbolic Logic 2:43. Church, A. 1940. On the concept of a random sequence. Bulletin of the American Mathematical Society 46:130–135. Church, A.

A (m* £ n* ) , and 2. A ¬ (n* ≤ m* ). 6 implies (1). 1, A ¬ (n* = 0). 7 A ¬ (n* ≤ 0), so (2) also holds. Now assume the result known for m = k and all n, and consider m = k + 1 < n. By induction hypothesis, A ¬ (n* ≤ k * ). 8, A [(n* ≤ (k + 1)* ) ⊃ [(n* £ k * ) ∨ (n* = (k + 1)* )]]. 1, A ¬ (n* £ (k + 1)* ), which proves (2). 9, A [(n* ≤ (k + 1)* ) ∨ ((k + 1)* ≤ n* )], which yields (1). 8 For each m ∈ N, we have A (" x )[( x £ m* ) ⊃ [( x = 0* ) ∨ ( x = 1* ) ∨ … ∨ ( x = m* )]]. 7. Assume the result known for m = k.

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