Today let’s talk about consistency.
Suppose I show you a map of Nebraska, with as-the-crow-flies distances marked between the major cities. Omaha to Lincoln, 100 miles. Lincoln to Grand Island, 100 miles. Omaha to Grand Island, 400 miles.
You are entitled to say “Hey, wait a minute! This map is inconsistent. The numbers don’t add up. If it’s 400 miles straight from Omaha to Grand Island, then there can’t be a 200 mile route that goes through Lincoln!”
So a map can be inconsistent. (It can also be consistent but wrong.) Nebraska itself, however, can no more be inconsistent than the color red can be made of terrycloth. (Red things can be made of terrycloth, but the color red certainly can’t.)
With that in mind, suppose I give you a theory of the natural numbers — that is, a list of axioms about them. You might examine my axioms and say “Hey! These axioms are inconsistent. I can use them to prove that 0 equals 1 and I can also use them to prove that 0 does not equal 1!” And, depending on the theory I gave you, you might be right. So a theory can be inconsistent. But the intended model of that theory — the natural numbers themselves — can no more be inconsistent than Nebraska can. Inconsistency in this context applies to theories, like the Peano axioms for arithmetic, not to structures, like the natural numbers themselves.
(See yesterday’s post for more on theories and models.)
Philosopher Alfred Korzybski admonishes us to remember that the map is not the territory. The theory is the map. The model is the territory. The hallmark of Internet crankery in this area is the refusal to distinguish them.
Whenever someone drones on at length about “the consistency (or inconsistency, or possible consistency, or possible inconsistency) of the natural numbers”, you’ll know he’s blathering. The concept simply doesn’t apply. Nebraska can’t be inconsistent. Only a description of it can be inconsistent.
(This brief snarky detour is brought to you by the small but determined band of commenters who consistently and vocally ignore this distinction in order to spout nonsense both here and on other blogs. I am not talking about anyone who’s commented here lately.)
It can be easy — and therefore entirely excusable — to get confused about this issue because in informal discussions of this subject — as in every other informal discussion of every subject in the English language — a single word can have multiple meanings. That sometimes happens with the word “arithmetic”. The phrase “Peano arithmetic” is the name of a theory — a list of axioms. On the other hand, some of us (me, for example) sometimes use the word “arithmetic” (a bit sloppily) to refer to a structure, namely the natural numbers themselves, which form a model of Peano arithmetic. Fortunately, the meaning is usually clear from context. If someone talks about “the consistency of arithmetic” you know that he’s talking about the theory (unless of course you have reason to suspect that he’s badly confused).
Now then. Let’s start with a theory. There are (at least) two sorts of questions you could ask about this theory. First: Is this theory consistent? In other words, is the theory free of self-contradiction? Second: Does this theory have a model? In other words, is there actually some structure that this theory describes?
If you’re given a map, the first question is like asking whether all the distances add up. The second question asks whether this is a map of someplace that actually exists or just a figment of the mapmaker’s imagination.
Inconsistent theories, obviously, have no models. A map that makes no sense cannot be a map of anything.
What about consistent theories? A consistent theory might a priori have either no models, or just one model, or many models.
The first possiblity is ruled out by Godel’s Completeness Theorem, not to be confused with the far more famous Godel’s Incompleteness Theorem. According to Godel’s Completeness Theorem, every consistent theory has at least one model. This is like saying that if you draw a map, and if nothing about the map is self-contradictory, then somewhere there is a territory that corresponds to the map. You should find this at least mildly surprising, but there it is.
If your theory is a theory of the natural numbers — in other words, if the natural numbers constitute a model for your theory — then the Lowenheim/Skolem Theorem says that your theory has a jillion other models as well. In other words, your map applies equally well to a jillion different territories. And there is no way, just by looking at the map, to tell those territories apart.
In other words, no theory — no list of axioms — can be a complete description of the natural numbers. It will always be a partial description, which applies equally well to a jillion other mathematical structures that look a little bit like the natural numbers but mostly a whole lot different.
Today’s moral: The map is not the territory. The map — the set of axioms — is either consistent or it’s not. If it’s inconsistent, there’s no corresponding territory. If it’s consistent, there are many corresponding territories and the map can’t tell you which one you’re in. That’s a fundamental limiitation on the power of the axiomatic method to describe a mathematical structure such as the natural numbers. It means there’s more to the natural numbers than any set of axioms can possibly know about.
Still to come: Are the Peano axioms consistent? Do the natural numbers really exist? And just how much about the natural numbers is any axiomatic system doomed not to know?
And finally: Thanks to those of you who encouraged me to continue this series. Let me know if you want still more.