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Systems of Logic Based on Ordinals was the PhD dissertation of the mathematician Alan Turing. [1] [2]Turing's thesis is not about a new type of formal logic, nor was he interested in so-called "ranked logic" systems derived from ordinal or relative numbering, in which comparisons can be made between truth-states on the basis of relative veracity.
Description numbers play a key role in many undecidability proofs, such as the proof that the halting problem is undecidable.In the first place, the existence of this direct correspondence between natural numbers and Turing machines shows that the set of all Turing machines is denumerable, and since the set of all partial functions is uncountably infinite, there must certainly be many ...
Alan Turing was driven to a terrible despair and early death by the nation he'd done so much to save. This remains a shame on the British government and British history. A pardon can go some way to healing this damage. It may act as an apology to many of the other gay men, not as well-known as Alan Turing, who were subjected to these laws. [193]
He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the Entscheidungsproblem ("decision problem"), the Frege–Church ontology, and the Church–Rosser theorem. Alongside his doctoral student Alan Turing, Church is considered one of the founders of computer science. [3] [4]
An earlier version of this thesis for classical computers was stated by Alan Turing's friend and student Robin Gandy in 1980. [2] [3] A similar thesis was stated by Michael Freedman in an early review of topological quantum computing with Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang, known as the Freedman-Church-Turing thesis: [4]
In computer science, a universal Turing machine (UTM) is a Turing machine capable of computing any computable sequence, [1] as described by Alan Turing in his seminal paper "On Computable Numbers, with an Application to the Entscheidungsproblem". Common sense might say that a universal machine is impossible, but Turing proves that it is possible.
The Church–Turing Thesis: Stephen Kleene, in Introduction To Metamathematics, finally goes on to formally name "Church's Thesis" and "Turing's Thesis", using his theory of recursive realizability. Kleene having switched from presenting his work in the terminology of Church-Kleene lambda definability, to that of Gödel-Kleene recursiveness ...
Jack Copeland is the Director of the Turing Archive for the History of Computing, [6] an extensive online archive on the computing pioneer Alan Turing. He has also written and edited books on Turing. He is one of the people responsible for identifying the concept of hypercomputation and machines more capable than Turing machines.