The study of the class of computable partial functions (i.e., recursive partial functions)
stands at the intersection of three fields: mathematics, theoretical computer science,
Mathematically, computability theory originates from the concept of an algorithm.
It leads to a classification of functions according their inherent complexity.
For the computer scientist, computability theory shows that quite apart from practical
matters of running time and memory space, there is a purely theoretical limit
to what computer programs can do. This is an important fact, and leads to the questions:
Where is the limit? What is on this side of the limit, and what lies beyond it?
Computability is relevant to the philosophy of mathematics and, in particular, to
the questions: What is a proof? Does every true sentence have a proof?
Computability theory is not an ancient branch of mathematics; it started in 1936. In
that year, Alonzo Church, Alan Turing, and Emil Post each published fundamental
papers that characterized the class of computable partial functions. Church’s article
introduced what is now called “Church’s thesis” (or the Church–Turing thesis), to
be discussed in Chapter 1. Turing’s article introduced what are now called “Turing
machines.” (1936 was also the year in which The Journal of Symbolic Logic began
publication, under the leadership of Alonzo Church and others. Finally, it was also the
year in which I was born.)