Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_
Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_
Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_
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So far, <strong>the</strong> system <strong>of</strong> Godel numbering as we have described it only sets up a<br />
correspondence between numbers <strong>and</strong> symbols, sequences <strong>of</strong> symbols, <strong>and</strong> sequences <strong>of</strong><br />
sequences <strong>of</strong> symbols <strong>of</strong> FA. But <strong>the</strong>re is more to <strong>the</strong> syntax <strong>of</strong> FA than this. There is also<br />
<strong>the</strong> question whe<strong>the</strong>r a given formula <strong>of</strong> FA is a well-formed formula, <strong>and</strong>, crucially, <strong>the</strong><br />
question whe<strong>the</strong>r a sequence <strong>of</strong> formulas constitutes a pro<strong>of</strong>. What Godel proved is that all<br />
<strong>the</strong> crucial functions needed to describe <strong>the</strong> complete syntax <strong>of</strong> FA, including being a wellformed<br />
formula <strong>and</strong> being a pro<strong>of</strong> <strong>of</strong> FA, corresponded to certain recursive functions in IA.<br />
A recursive function is one that, intuitively speaking, can be mechanically computed. This<br />
kind <strong>of</strong> function can also be characterized strictly ma<strong>the</strong>matically, <strong>and</strong> this Godel<br />
proceeded to do. An example <strong>of</strong> a so-called primitive recursive function, <strong>the</strong> "+ function,"<br />
also known as <strong>the</strong> "addition function," will illustrate what is meant by recursivity. Let us<br />
call <strong>the</strong> number that comes right after a natural number x <strong>the</strong> successor <strong>of</strong> x, or s(x). The<br />
"+" function, <strong>the</strong>n, is given by two rules:<br />
(a) x + 0 = x;<br />
(b) x + s(y) = s(x + y). (This can be read aloud as "x plus <strong>the</strong> successor <strong>of</strong> y equals <strong>the</strong><br />
successor <strong>of</strong> x-plus-y.")<br />
The successor <strong>of</strong> x, namely, s(x), can be defined as x + 1. This kind <strong>of</strong> recursive definition<br />
can be used to compute mechanically, by a<br />
kind <strong>of</strong> "bootstrapping," <strong>the</strong> sum <strong>of</strong> any two natural numbers, since every natural number is<br />
ei<strong>the</strong>r 0 or <strong>the</strong> successor <strong>of</strong> some o<strong>the</strong>r natural number.<br />
Recursive definitions were studied by Dedekind, Peano, Skolem <strong>and</strong> o<strong>the</strong>rs, <strong>and</strong> recursive<br />
functions had been used implicitly throughout <strong>the</strong> history <strong>of</strong> ma<strong>the</strong>matics, but <strong>the</strong> first to<br />
elaborate a precise <strong>and</strong> forceful account <strong>of</strong> such functions was Godel, who cited his young<br />
French colleague Jacques Herbr<strong>and</strong> as having influenced his underst<strong>and</strong>ing <strong>of</strong> <strong>the</strong>se ideas.<br />
Herbr<strong>and</strong> had written to Godel on hearing <strong>of</strong> his incompleteness results from Von<br />
Neumann. Godel wrote a detailed <strong>and</strong> deeply respectful response, at <strong>the</strong> end <strong>of</strong> which he<br />
suggested that in <strong>the</strong> future <strong>the</strong>y correspond each in his mo<strong>the</strong>r tongue. (Godel was very<br />
good at languages.) He never received a reply. What he did receive was a touching letter<br />
from Herbr<strong>and</strong>'s fa<strong>the</strong>r informing him that <strong>the</strong> reason for his son's silence was that he had<br />
fallen to his death while climbing in <strong>the</strong> Alps. Jacques Herbr<strong>and</strong> had been just twentythree<br />
years old.<br />
Godel demonstrated, <strong>the</strong>n, that <strong>the</strong> fundamental concepts <strong>of</strong> MFA, in which was found <strong>the</strong><br />
metama<strong>the</strong>matics, or pro<strong>of</strong> <strong>the</strong>ory, <strong>of</strong> FA, corresponded to certain recursive functions in<br />
IA. In particular, <strong>the</strong> function Bew(x, y), i.e., x is a pro<strong>of</strong> <strong>of</strong> y (from <strong>the</strong> German for pro<strong>of</strong>,<br />
Beweis), when coded into natural numbers, yields a recursive function. This was important<br />
because in proving that FA can represent IA, Godel had already shown that any recursive<br />
function contained in IA could be represented in FA. Specifically, if <strong>the</strong>re was a truth<br />
about a recursive function in IA, <strong>the</strong>re would be a corresponding formula that was a<br />
<strong>the</strong>orem <strong>of</strong> FA. Once he had demonstrated that <strong>the</strong> basic functions in MFA when coded into<br />
natural numbers yield recursive functions, he could conclude that MFA, just like IA, could