Douglas Hofstadter - I Am a Strange Loop

For multi-symbol formulas (by the way, in this book the terms “string of symbols” — “string” for short — and “formula” are synonymous), the idea was to replace the symbols, one by one, moving left to right, by their code numbers, and then to combine all of those individual code numbers (by using them as exponents to which successive prime numbers are raised) into one unique big integer. Thus, once isolated symbols had been assigned numbers, the numbers assigned to strings of symbols were not arbitrary.

For instance, suppose that the (arbitrary) code number for the symbol “0” is 2, and the code number for the symbol “=” is 6. Then for the three symbols in the very simple formula “0=0”, the code numbers are 2, 6, 2, and these three numbers are used as exponents for the first three prime numbers (2, 3, and 5) as follows:

2 2 · 3 6 · 5 2 = 72900

So we see that 72900 is the single number that corresponds to the formula “0=0”. Of course this is a rather large integer for such a short formula, and you can easily imagine that the integer corresponding to a fifty-symbol formula is astronomical, since it involves putting the first fifty prime numbers to various powers and then multiplying all those big numbers together, to make a true colossus. But no matter — numbers are just numbers, no matter how big they are. (Luckily for Gödel, there are infinitely many primes, since if there had been merely, say, one billion of them, then his method would only have let him encode formulas made of a billion symbols or fewer. Now that would be a crying shame!)

The decoding process works by finding the prime factorization of 72900 (which is unique), and reading off the exponents that the ascending primes are raised to, one by one — 2, 6, 2 in this case.

To summarize, then, in this non-obvious but simple manner, Gödel had found a way to replace any given formula of PM by an equivalent number (which other people soon would dub its Gödel number ). He then extended this idea of “arithmetization” to cover arbitrary sequences of formulas, since proofs in PM are sequences of formulas, and he wanted to be able to deal with proofs, not just isolated formulas. Thus an arbitrarily long sequence of formulas could be converted into one large integer via essentially the same technique, using primes and exponents. You can imagine that we’re talking really big numbers here.

In short, Gödel showed how any visual symbol-pattern whatsoever in the idiosyncratic notation of Principia Mathematica could be assigned a unique number, which could easily be decoded to give back the visual pattern ( i.e., sequence of symbols) to which it corresponded. Conceiving of and polishing this precise two-way mapping, now universally called “Gödel numbering”, constituted the first key step of Gödel’s work.

Very Big Integers Moving in Lock-step with Formulas

The next key step was to make Fibonacci-like recursive definitions of special sets of integers — integers that would organically grow out of previously generated ones by addition or multiplication or more complex computations. One example would be the wff numbers, which are those integers that, via Gödel’s code, represent “well-formed” or “meaningful” formulas of PM, as opposed to those that represent meaningless or ungrammatical strings. (A sample well-formed formula, or “wff ” for short, would be “0+0=sss0”. Though it asserts a falsity, it’s still a meaningful statement. On the other hand, “=)0(=” and “00==0+=” are not wffs. Like the arbitrary sequence of pseudo-words “zzip dubbiwubbi pizz”, they don’t assert anything.) Since, as it happens, longer wffs are built up in PM from shorter wffs by just a few simple and standard rules of typographical juxtaposition, their larger code numbers can likewise be built up from the smaller code numbers of shorter ones by just a few simple and standard rules of numerical calculation.

I’ve said the foregoing rather casually, but in fact this step was perhaps the deepest of Gödel’s key insights — namely, that once strings of symbols had been “arithmetized” (given numerical counterparts), then any kind of rule-based typographical shunting-around of strings on paper could be perfectly paralleled by some kind of purely arithmetical calculation involving their numerical proxies — which were huge numbers, to be sure, but still just numbers. What to Russell and Whitehead looked like elaborate symbol-shunting looked like a lot of straightforward number-crunching to Kurt Gödel (although of course he didn’t use that colorful modern term, since this was all taking place back in the prehistoric days when computers didn’t yet exist). These were simply two different views of what was going on — views that were 100 percent equivalent and interchangeable.

Glimmerings of How PM Can Twist Around and See Itself

Gödel saw that the game of building up an infinite class of numbers, such as wff numbers, through recursion — that is, making new “members of the club” by combining older, established members via some number-crunching rule — is essentially the same idea as Fibonacci’s recursive game of building up the class of F numbers by taking sums of earlier members. Of course recursive processes can be far more complicated than just taking the sum of the latest two members of the club.

What a recursive definition does, albeit implicitly, is to divide the entire set of integers into members and non-members of the club — that is, those numbers that are reachable, sooner or later, via the recursive building-up process, and those that are never reachable, no matter how long one waits. Thus 34 is a member of the F club, whereas 35 is a non-member. How do we know 35 is not an F number? That’s very easy — the rule that makes new F numbers always makes larger ones from smaller ones, and so once we’ve passed a certain size, there’s no chance we’ll be returning to “pick up” other numbers in that vicinity later. In other words, once we’ve made the F numbers 1, 2, 3, 5, 8, 13, 21, 34, 55, we know they are the only ones in that range, so obviously 35, 36, and so on, up to 54, are not F numbers.

If, however, some other club of numbers is defined by a recursive rule whose outputs are sometimes bigger than its inputs and other times are smaller than its inputs, then, in contrast to the simple case of the F club, you can’t be so sure that you won’t ever be coming back and picking up smaller integers that were missed in earlier passes.

Let’s think a little bit more about the recursively defined club of numbers that we called “wff numbers”. We’ve seen that the number 72900 possesses “wff-ness”, and if you think about it, you can see that 576 and 2916 lack that quality. (Why? Well, if you factor them and look at the exponents of 2 and 3, you will see that these numbers are the numerical encodings of the strings “0=” and “=0”, respectively, neither of which makes sense, whence they are not well-formed formulas.) In other words, despite its odd definition, wff-ness, no more and no less than squareness or primeness or Fibonacci’s F -ness, is a valid object of study in the world of pure number. The distinction between members and non-members of the “wff club” is every bit as genuine a number-theoretical distinction as that between members and non-members of the club of squares, the club of prime numbers, or the club of F numbers, for wff numbers are definable in a recursive arithmetical ( i.e., computational) fashion. Moreover, it happens that the recursive rules defining wff-ness always produce outputs that are bigger than their inputs, so that wff-ness shares with F -ness the simple property that once you’ve exceeded a certain magnitude, you know you’ll never be back visiting that zone again.