10.05.2014 Views

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_

SHOW MORE
SHOW LESS

You also want an ePaper? Increase the reach of your titles

YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.

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

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!