[[File:Women holding parts of the first four Army computers.jpg|right|thumb|Parts from four early computers, 1962. From left to right: [[ENIAC]] board, [[EDVAC]] board, [[ORDVAC]] board, and [[BRLESC]]-I board, showing the trend toward [[miniaturization]].]]
The principle of the modern computer was first described by [[computer scientist]] [[Alan Turing]], who set out the idea in his seminal 1936 paper,<ref name=Turing-1937-1938>{{harvs|nb |last=Turing |year=1937 |year2=1938}}</ref> ''On Computable Numbers''. Turing reformulated [[Kurt Gödel]]'s 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as [[Turing machine]]s. He proved that some such machine would be capable of performing any conceivable mathematical computation if it were representable as an [[algorithm]]. He went on to prove that there was no solution to the ''[[Entscheidungsproblem]]'' by first showing that the [[halting problem]] for Turing machines is [[Decision problem|undecidable]]: in general, it is not possible to decide algorithmically whether a given Turing machine will ever halt.
He also introduced the notion of a "universal machine" (now known as a [[universal Turing machine]]), with the idea that such a machine could perform the tasks of any other machine, or in other words, it is provably capable of computing anything that is computable by executing a program stored on tape, allowing the machine to be programmable. [[John von Neumann|Von Neumann]] acknowledged that the central concept of the modern computer was due to this paper.<ref>{{harvnb|Copeland|2004|p=22}}: "von Neumann ... firmly emphasized to me, and to others I am sure, that the fundamental conception is owing to Turing—insofar as not anticipated by Babbage, Lovelace and others. Letter by [[Stanley Frankel]] to [[Brian Randell]], 1972."</ref> Turing machines are to this day a central object of study in [[theory of computation]]. Except for the limitations imposed by their finite memory stores, modern computers are said to be [[Turing-complete]], which is to say, they have [[algorithm]] execution capability equivalent to a [[universal Turing machine]].
