Algorithm
The word "algorithm" itself derives from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized "Algoritmi". A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem (decision problem) posed by David Hilbert in 1928. Later formalizations were framed as attempts to define "effective calculability" or "effective method". Those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, and Alan Turing's Turing machines of 1936–37 and 1939.
Algorithm
An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus, Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an "arbitrary" "input" integer or integers that, in theory, can be arbitrarily large. Thus an algorithm can be an algebraic equation such as "y = m + n" – two arbitrary "input variables" "m" and "n" that produce an output "y". But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example):