Editing Hypercomputation (section)

==History==
A computational model going beyond Turing machines was introduced by [[Alan Turing]] in his 1938 PhD dissertation ''[[Systems of Logic Based on Ordinals]]''.<ref>{{Cite journal |doi = 10.1112/plms/s2-45.1.161|title = Systems of Logic Based on Ordinals†|year = 1939|last1 = Turing|first1 = A. M.|journal = Proceedings of the London Mathematical Society|volume = 45|pages = 161–228|hdl = 21.11116/0000-0001-91CE-3|hdl-access=free}}</ref> This paper investigated mathematical systems in which an [[Oracle machine|oracle]] was available, which could compute a single arbitrary (non-recursive) function from [[Natural number|naturals]] to naturals. He used this device to prove that even in those more powerful systems, [[Undecidable problem|undecidability]] is still present. Turing's oracle machines are mathematical abstractions, and are not physically realizable.<ref>"Let us suppose that we are supplied with some unspecified means of solving number-theoretic problems; a kind of oracle as it were. We shall not go any further into the nature of this oracle apart from saying that it cannot be a machine" (Undecidable p.&nbsp;167, a reprint of Turing's paper ''Systems of Logic Based On Ordinals'')</ref>