Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_

So far, the system of Godel numbering as we have described it only sets up a
correspondence between numbers and symbols, sequences of symbols, and sequences of
sequences of symbols of FA. But there is more to the syntax of FA than this. There is also
the question whether a given formula of FA is a well-formed formula, and, crucially, the
question whether a sequence of formulas constitutes a proof. What Godel proved is that all
the crucial functions needed to describe the complete syntax of FA, including being a wellformed
formula and being a proof of FA, corresponded to certain recursive functions in IA.
A recursive function is one that, intuitively speaking, can be mechanically computed. This
kind of function can also be characterized strictly mathematically, and this Godel
proceeded to do. An example of a so-called primitive recursive function, the "+ function,"
also known as the "addition function," will illustrate what is meant by recursivity. Let us
call the number that comes right after a natural number x the successor of x, or s(x). The
"+" function, then, is given by two rules:
(a) x + 0 = x;
(b) x + s(y) = s(x + y). (This can be read aloud as "x plus the successor of y equals the
successor of x-plus-y.")
The successor of x, namely, s(x), can be defined as x + 1. This kind of recursive definition
can be used to compute mechanically, by a
kind of "bootstrapping," the sum of any two natural numbers, since every natural number is
either 0 or the successor of some other natural number.
Recursive definitions were studied by Dedekind, Peano, Skolem and others, and recursive
functions had been used implicitly throughout the history of mathematics, but the first to
elaborate a precise and forceful account of such functions was Godel, who cited his young
French colleague Jacques Herbrand as having influenced his understanding of these ideas.
Herbrand had written to Godel on hearing of his incompleteness results from Von
Neumann. Godel wrote a detailed and deeply respectful response, at the end of which he
suggested that in the future they correspond each in his mother tongue. (Godel was very
good at languages.) He never received a reply. What he did receive was a touching letter
from Herbrand's father informing him that the reason for his son's silence was that he had
fallen to his death while climbing in the Alps. Jacques Herbrand had been just twentythree
years old.
Godel demonstrated, then, that the fundamental concepts of MFA, in which was found the
metamathematics, or proof theory, of FA, corresponded to certain recursive functions in
IA. In particular, the function Bew(x, y), i.e., x is a proof of y (from the German for proof,
Beweis), when coded into natural numbers, yields a recursive function. This was important
because in proving that FA can represent IA, Godel had already shown that any recursive
function contained in IA could be represented in FA. Specifically, if there was a truth
about a recursive function in IA, there would be a corresponding formula that was a
theorem of FA. Once he had demonstrated that the basic functions in MFA when coded into
natural numbers yield recursive functions, he could conclude that MFA, just like IA, could