Despite being bagel-deprived, ancient Greek philosophers posed deep questions and pondered partial answers in ways that still deserve attention. In particular, one of them showed how slippery is the notion of truth by considering the Liar Paradox:
This statement is not true.
To give a short name to the statement just quoted, I will call it LP. If LP is true, then it is false, which implies that it is true, and so on. It is like a puppy chasing its tail. Very annoying to anybody who agrees with what another ancient Greek philosopher said about the excluded middle. Yes, there many utterances that are not statements and so are neither true nor false. But the duck test says that LP is a statement.
For millenia, the usual way to dodge the Liar Paradox has been to abstain from self-referential language. Tho I may talk about myself, I never talk about myself talking about myself. Except in previous sentence. Avoiding LP-ish talk is harder than just avoiding the exact words of LP itself. How much harder? Apart from some temporary scares that I will ignore in this post, everybody thought it was only a little harder. Until 1931.
Suppose we are willing to pause in our search for all the answers to all the questions; we restrict our attention to some narrow subject and restrict our language to statements about that subject. We can agree to talk only about bagels for a while. We want to use language expressive enough to say interesting things about bagels, but not so expressive that we blunder into LP land. Maybe, if we are both careful and lucky, we can console ourselves for talking about so little by celebrating success in saying so much about it. Maybe we can discover a Grand Unified Theory of Bagels that encompasses the truth, the whole truth, and nothing but the truth (so long as we speak in a restricted language that can still say a lot about bagels). Welcome to bagelology.
Suppose we have already agreed on the syntax and semantics of statements in bagelology. What next? We might make a list of some important things we already know about bagels. For now, it does not matter whether we got this knowledge from empirical observation or divine revelation. I will call these statements the “axioms” of bagelology. The axioms are all well and good, but we want new knowledge. One way to get new knowledge is by logical deduction, which ancient Greeks also pioneered. (No, they did not do it all.) So we get up to speed about rules of inference and agree that whatever can deduced from the axioms is a “theorem” of bagelology. Logic has its pros and cons; a big pro is the way we can keep adding to whatever we have. We can toss yesterday’s theorems in with the axioms and bake (i.e., deduce) more theorems today, along with some bagels.
After baking a new batch of theorems every day for many days, we have a substantial body of knowledge. Of course it is still incomplete. Somebody writes down a statement about bagels and asks if it is true. Dunno. Neither the statement nor its negation appears in our growing inventory of theorems that have already been baked. Maybe the next batch of theorems will answer the question. Maybe not. We can still hope that our Grand Unified Theory of Bagels is “complete” in the weaker sense that someday the answer will appear. Just keep on baking.
There is not much to be said about bagels without talking about flour. Come to think of it, we also need to count bagels and price them. We need to talk about numbers. Nothing fancy; just plain old integer arithmetic. Throw it all in. Bagels; flour; ovens; numbers — we still have a shot at a complete theory, which eventually decides whether any given bagelogical statement is true or false.
Oops. The greatest logician in history was not an ancient Greek. He was Kurt Gödel, who showed that bagelology (and anything else that uses the axiom/theorem paradigm) cannot be both complete and expressive enough to support arithmetic. Yes, he dropped that bomb in 1931, long before we wrote down our axioms. He did not need to know them. He knew that we would be in trouble as soon as he knew that we would count and calculate. Arithmetic is a Trojan horse.
There is still a trivial way to make every true bagelogical statement appear eventually in a batch of theorems. An inconsistent theory that just asserts everything will do the job. Every statement appears eventually, and each true statement may appear either before or after the false one that denies it. An inconsistent theory tells us nothing and is useless, except in politics.
No matter how cleverly we set up bagelology, there are true statements that are not theorems, unless bagelology is either inconsistent or too weak to support arithmetic. Really. Gödel proved this with a combination of deep insight, technical virtuousity, and a little help from an ancient Greek. He showed how to seduce bagelology into talking about itself implicitly (despite that pledge of abstinence from self-reference) and then into saying something much like LP:
This statement is not a theorem in this system.
Having shed a few tears for completeness and saluted great thinkers (both ancient and modern), we can go on living. If we do set up bagelology well, it may be consistent and informative, revealing much that is true (and maybe also good and beautiful) but not at all obvious from a glance at the axioms. There will always be truths that are not theorems, and we may sometimes discover a few of them by other means. We can advance bagelology by adding those discoveries to the older axioms; we should never declare victory and carve the current set of axioms in stone.
BTW, I know there is a difference between supporting arithmetic and just using some of it as a black box. If U know enough mathematical logic to exploit that difference in a Houdini escape from incompleteness for bagelology, then I salute U and hope U still enjoyed my whimsical way of summarizing one of Gödel’s discoveries for those who do not already know and admire it.
Now it is time for something that really is about bagels. If U believed what U read on package labels and in food columns, U can improve your own version of bagelology by replacing that old axiom about storing things at room temperature (or in a cool, dry place) with Rosen’s Rule:
It is OK to refrigerate bagels. Likewise for bread.
Once a bagel is no longer warm from the oven, it needs to be toasted before being eaten. A bagel pulled from the frige recently takes a little longer to toast; the result is ever so slightly chewier than with room temperature storage. More than a week after purchase, a wrapped bagel or bagged loaf of bread in the frige is still good. Tho it may not be quite so good as when first brought home, it does not sport the green fuzz that adorns the poor schmuck stored the conventional way for more than a few days.
A small household that dislikes green fuzz can still buy baked goods in convenient quantities w/o being forced to pig out or throw out. While the toaster does its job, U can read a haiku that talks about itself:
Basho Meets GödelHaiku written on a typewriter: ultimate incongruity.