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Imagine a Turing machine that is started on a completely blank tape, and eventually halts.
Let the notation \(P(n)\downarrow\) mean that Turing machine \(P\) on input \(n\) eventually halts.
Given a program, it is in general not possible to figure out what it does and whether or not it eventually halts.
Until Andrew Wiles relatively recently proved Fermat's Last Theorem, all the mathematicians in the world, working for over three centuries, were unable to decide whether or not \(M_{F}\) on input "0" eventually halts.
If we start the Turing machine \(M\) on a tape containing \(n\) and it eventually halts with its tape containing \(m\), then we say that \(M\) on input \(n\), computes \(m: M n) = m\).
For a Turing machine, \(P\), define \(L(P \), the set accepted by \(P\), to be those numbers \(n\) such that \(P\) on input \(n\) eventually halts, The above argument shows that if a set \(S\) is r.e. then it is accepted by some Turing machine, \(P\), i.e., \(S = L(P \).
Similar(53)
It was eventually halted.
Some of these canals were constructed, although canal construction was eventually halted for other reasons.
The scrum-half raced upfield and although the attack was eventually halted, the defence were stretched.
Reducing the concentration of CO2 would slow and perhaps eventually halt the production of carbonic acid, which in turn would reduce ocean acidification.
"To stabilise the climate and eventually halt the rising trend of extreme heat, carbon emissions need to be cut rapidly and deeply," the report said.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com