I Am a Strange Loop (30 page)

Well, suppose you wanted to know if statement X is true or false (for instance, the famous claim “Every even number greater than 2 is the sum of two primes” — which, as I stated above, remains unsettled even today, after nearly three centuries of work). You would just write X down in the formal notation of
PM,
then convert that formula mechanically into its Gödel number
x,
and feed that number into Göru (thus asking if
x
is prim or not). Of course
x
will be a huge integer, so it would probably take Göru a good while to give you an answer, but (assuming that Göru is not a hoax) sooner or later it would spit out either a “yes” or a “no”. In case Göru said “yes”, you would know that
x
is a prim number, which tells you that the formula it encodes is a
provable
formula, which means that statement X is true. Conversely, were Göru to tell you “no”, then you would know that the statement X is
not
provable, and so, believing in the Mathematician’s Credo (
Principia Mathematica
version), you would conclude it is false.

In other words, if we only had a machine that could infallibly tell apart prim numbers and “saucy” (non-prim) numbers, and taking for granted that the
Principia Mathematica
version of the Mathematician’s Credo is valid, then we could infallibly tell true statements from false ones. In short, having a Göru would give us a royal key to all of mathematical knowledge.

The prim numbers alone would therefore seem to contain, in a cloaked fashion,
all of mathematical knowledge
wrapped up inside them! No other sequence of numbers ever dreamt up by anyone before Gödel had anything like this kind of magically oracular quality. These amazing numbers seem to be worth their weight in gold! But as I told you, the prim numbers are elusive, because small ones sometimes wind up being added to the club at very late stages, so it won’t be easy to tell prim numbers from saucy ones, nor to build a Göru. (This is meant as a premonition of things to come.)

Gödelian Strangeness

Finally, Gödel carried his analogy to its inevitable, momentous conclusion, which was to spell out for his readers (not symbol by symbol, of course, but via a precise set of “assembly instructions”) an astronomically long formula of
PM
that made the seemingly innocent assertion, “A certain integer
g
is not a prim number.” However, that “certain integer
g
” about which this formula spoke happened, by a most unaccidental (some might say diabolical) coincidence, to be the number associated with (
i.e.,
coding for)
this very formula
(and so it was necessarily a gargantuan integer). As we are about to see, Gödel’s odd formula can be interpreted on two different levels, and it has two very different meanings, depending on how one interprets it.

On its more straightforward level, Gödel’s formula merely asserts that this gargantuan integer
g
lacks the number-theoretical property called
primness.
This claim is very similar to the assertion “72900 is not a prime number”, although, to be sure,
g
is a lot larger than 72900, and primness is a far pricklier property than is primeness. However, since primness was defined by Gödel in such a way that it numerically mirrored the provability of strings via the rules of the
PM
system, the formula
also
claims:

The formula that happens to have the code number
g
is not provable via the rules of
Principia Mathematica.

Now as I already said, the formula that “just happens” to have the code number
g
is the formula making the above claim. In short, Gödel’s formula is making a claim about
itself
— namely, the following claim:

This very formula is not provable via the rules of
PM.

Sometimes this second phraseology is pointedly rendered as “I am not a theorem” or, even more tersely, as

I am unprovable

(where “in the
PM
system” is tacitly understood).

Gödel further showed that his formula, though very strange and discombobulating at first sight, was not all that unusual; indeed, it was merely one member of an infinite family of formulas that made claims about the system
PM,
many of which asserted (some truthfully, others falsely) similarly weird and twisty things about themselves (
e.g.,
“Neither I nor my negation is a theorem of
PM
”, “If I have a proof inside
PM,
then my negation has an even shorter proof than I do”, and so forth and so on).

Young Kurt Gödel — he was only 25 in 1931 — had discovered a vast sea of amazingly unsuspected, bizarrely twisty formulas hidden inside the austere, formal, type-theory-protected and therefore supposedly paradoxfree world defined by Russell and Whitehead in their grandiose threevolume œuvre
Principia Mathematica,
and the many counterintuitive properties of Gödel’s original formula and its countless cousins have occupied mathematicians, logicians, and philosophers ever since.

How to Stick a Formula’s Gödel Number inside the Formula

I cannot leave the topic of Gödel’s magnificent achievement without going into one slightly technical issue, because if I failed to do so, some readers would surely be left with a feeling of confusion and perhaps even skepticism about a key aspect of Gödel’s work. Moreover, this idea is actually rather magical, so it’s worth mentioning briefly.

The nagging question is this: How on earth could Gödel fit a formula’s Gödel number into the formula itself? When you think about it at first, it seems like trying to squeeze an elephant into a matchbox — and in a way, that’s exactly right. No formula can literally contain the numeral for its own Gödel number, because that numeral will contain many more symbols than the formula does! It seems at first as if this might be a fatal stumbling block, but it turns out not to be — and if you think back to our discussion of G. G. Berry’s paradox, perhaps you can see why.

The trick involves the simple fact that some huge numbers have very short descriptions (387420489, for instance, can be described in just four syllables: “nine to the ninth”). If you have a very short recipe for calculating a very long formula’s Gödel number, then instead of describing that huge number in the most plodding, clunky way (“the successor of the successor of the successor of …… the successor of the successor of zero”), you can describe it via your computational shortcut, and if you express your shortcut in symbols (rather than inserting the numeral itself) inside the formula, then you can make the formula talk about itself without squeezing an elephant into a matchbox. I won’t try to explain this in a mathematical fashion, but instead I’ll give an elegant linguistic analogy, due to the philosopher W. V. O. Quine, which gets the gist of it across.

Gödel’s Elephant-in-Matchbox Trick via Quine’s Analogy

Suppose you wanted to write a sentence in English that talks about itself without using the phrase “this sentence”. You would probably find the challenge pretty tricky, because you’d have to actually
describe
the sentence inside itself, using quoted words and phrases. For example, consider this first (somewhat feeble) attempt:

The sentence “This sentence has five words” has five words.

Now what I’ve just written (and you’ve just read) is a sentence that is true, but unfortunately it’s not about itself. After all, the full thing contains
ten
words, as well as some quotation marks. This sentence is about a shorter sentence embedded inside it, in quote marks. And changing “five” to “ten” still won’t make it refer to itself; all that this simple act does is to turn my sentence, which was true, into a false one. Take a look:

The sentence “This sentence has ten words” has ten words.

This sentence is false. And more importantly, it’s
still
merely about a shorter sentence embedded inside itself. As you see, so far we are not yet very close to having devised a sentence that talks about itself.

The problem is that anything I put inside quote marks will necessarily be shorter than the entire sentence of which it is a part. This is trivially obvious, and in fact it is an exact linguistic analogue to the stumbling block of trying to stick a formula’s own Gödel number directly inside the formula itself. An elephant will not fit inside a matchbox! On the other hand, an elephant’s DNA
will
easily fit inside a matchbox…

And indeed, just as DNA is a
description
of an elephant rather than the elephant itself, so there is a way of getting around the obstacle by using a
description
of the huge number rather than the huge number itself. (To be slightly more precise, we can use a concise symbolic description instead of using a huge
numeral.
) Gödel discovered this trick, and although it is quite subtle, Quine’s analogy makes it fairly easy to understand. Look at the following sentence fragment, which I’ll call “Quine’s Quasi-Quip”:

preceded by itself in quote marks yields a full sentence.

As you will note, Quine’s Quasi-Quip is certainly
not
a full sentence, for it has no grammatical subject (that is, “yields” has no subject); that’s why I gave it the prefix “Quasi”. But what if we were to put a noun at the head of the Quasi-Quip — say, the title “Professor Quine”? Then Quine’s Quasi-Quip will turn into a full sentence, so I’ll call it “Quine’s Quip”:

“Professor Quine” preceded by itself in quote marks yields a full sentence.

Here, the verb “yields”
does
have a subject — namely, Professor Quine’s title, modified by a trailing adjectival phrase that is six words long.

But what does Quine’s Quip
mean
? In order to figure this out, we have to actually
construct
the entity that it’s talking about, which means we have to precede Professor Quine’s title by itself in quote marks. This gives us:

“Professor Quine” Professor Quine

The Quine’s Quip that we created a moment ago merely asserts (or rather, claims) that this somewhat silly phrase is a full sentence. Well, that claim is obviously false. The above phrase is
not
a full sentence; it doesn’t even contain a verb.

However, we arbitrarily used Professor Quine’s title when we could have used a million different things. Is there some
other
noun that we might place at the head of Quine’s Quasi-Quip that will make Quine’s Quip come out
true
? What Gödel realized, and what Quine’s analogy helps to make clear, is that for this to happen, you have to use, as your subject of the verb “yields”, a
subjectless sentence fragment.

What is an example of a subjectless sentence fragment? Well, just take any old sentence such as “Snow is white”, and cut off its subject. What you get is a subjectless sentence fragment: “is white”. So let’s use
this
as the noun to place in front of Quine’s Quasi-Quip:

“is white” preceded by itself in quote marks yields a full sentence.

This medium-sized mouthful makes a claim about a construction that we have yet to exhibit, and so let’s do so without further ado:

“is white” is white.

(I threw in the period for good measure, but let’s not quibble.)

Now what we have just produced certainly
is
a full sentence, because it has a verb (“is”), and that verb has a subject (the quoted phrase), and the whole thing makes sense. I’m not saying that it is
true,
mind you, for indeed it is blatantly false: “is white” is in fact
black
(although, to be fair, letters and words do contain some white space along with their black ink, otherwise we couldn’t read them). In any case, Quine’s Quasi-Quip when fed “is white” as its input yielded a full sentence, and that’s exactly what Quine’s Quip claimed. We’re definitely making headway.

The Trickiest Step

Our last devilish trick will be to use Quine’s Quasi-Quip
itself
as the noun to place at its head. Here, then, is Quine’s Quasi-Quip with a quoted copy of itself installed in front:

“preceded by itself in quote marks yields a full sentence”
preceded by itself in quote marks yields a full sentence.

What does this Quip claim? Well, first we have to determine what entity it is talking
about,
and that means we have to construct the analogue to “ ‘is white’ is white”. Well, in this case, the analogue is the following:

“preceded by itself in quote marks yields a full sentence”
preceded by itself in quote marks yields a full sentence.

I hope you are not lost at this point, for we really have hit the crux of the matter. Quine’s Quip turns out to be talking about a phrase that is identical to the Quip itself! It is claiming that
something
is a full sentence, and when you go about constructing that thing, it turns out to be Quine’s Quip itself. So Quine’s Quip talks about itself, claiming of itself that it is a full sentence (which it surely is, even though it is built out of two subjectless sentence fragments, one in quote marks and one not).

While you are pondering this, I will jump back to the source of it all, which was Gödel’s
PM
formula that talked about itself. The point is that Gödel numbers, since they can be used as
names
for formulas and can be
inserted
into formulas, are precisely analogous to quoted phrases. Now we have just seen that there is a way to use quotation marks and sentence fragments to make a full sentence that talks about itself (or if you prefer, a sentence that talks about
another
sentence, but one that is a clone to it, so that whatever is true of the one is true of the other).