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

Triple Fugue: Intuitive Mathematics, Formal Mathematics, and Metamathematics
Godel's beautiful fugue was constructed from three distinguishable mathematical
languages or theories. The beauty was to be found in the pattern of relationships woven
from the three parts. To begin with, there was intuitive arithmetic, the arithmetic found
in mathematical textbooks written in the language of ordinary mathematics. Call this
language or theory IA (for intuitive arithmetic). The propositions of IA are sentences with
content: they express truths or falsehoods about numbers. Next there was a formal
deductive system for arithmeticóin Godel's proof, a system of pure syntax put forward by
Bertrand Russell, modeled on the original by Fregeówith a specified set of axioms and
explicit rules of deduction that determined which formulas were theorems. Call this
system FA (for formal arithmetic). The "sentences" of FA are simply formulas without
semantic content. In themselves, they are neither true nor false. They are, however,
either provable from the axioms of FA or not. If provable, they are called theorems. FA,
however, is so designed that we can give it an interpretation, a semantics, under which it
can be read as corresponding to IA. That is, FA is designed to mirror IA, so that if all goes
well, there will be an exact one-to-one correspondence between the numerals in FA and
the numbers in IA, and a similar correspondence between the true sentences
of IA and the theorems or FA. Pur succinctly, FA is designed to represent IA.
The third language or theory is the metatheory of formal arithmetic, the framework in
which the syntactic rules, the proof theory, of FA is spelled out. Call this language MFA
(for the metatheory of formal arithmetic). If FA is the machine, MFA is the owner's manual
that specifies how the machine works. Like IA, it consists of meaningful sentences that
have truth values. MFA specifies which formulas of FA are "well-formed formulas," meaning
that they satisfy the official rules of formula construction. Crucially, it also specifies what
it means to be a proof in FA.
Godel's insight was to see that FA could be used to represent not only IAóto the extent to
which this is possibleóbut also MFA. He proved the latter by showing how MFA could be
represented in IA, via a revolutionary device known today as the "arithmetization of metamathematics."
But then if FA can represent IA, it can also, via the interpretation of MFA in
IA, represent MFA. That is, FA can represent its own metatheory. The trick, then, was to
construct a formula of FA that would have two simultaneous meanings in two languages,
MFA and IA. Godel was able to exhibit just such a formula and to prove that it was
simultaneously unprovable in FA and, intuitively, true in IA and MFA. This would be a
formula provably unprovable in FA, and yet expressing a true proposition in IA about the
natural numbers as well as a true proposition in MFA about its own unprovability.
Nothing like this had ever been seen before. Godel had skirted around the deadly liar's
paradox, substituting for it an unproblematic unprovability paradox (which was not really a
paradox at all); established the possibility and harmlessness of self-reference;
demonstrated representability relationships among three distinct languages; arithme-tized
the syntax of one of those languages; and finally, exhibited a formula of one language that
was provably unprovable and simultaneously true. This was logic, it was mathematics, but
it didn't look like logic or mathematics. It looked more like Kafka. Indeed, when the
mathematician Paul Cohen, a Fields medalist who proved the independence