Gödel’s incompleteness theorems are the kind of theorems that break your brain.

In the last post, we discussed the theorems themselves, and their consequences. In short, they show the inherent limitations of mathematics.

The first theorem relates two concepts: consistency and provability. A mathematical system (a set of assumptions which are called axioms) is consistent if there aren’t any contradictions. In other words, you can’t prove a statement both true and false.

Inside of any logical system, there are many statements, i.e., things you can say. I could say something like “All prime numbers are smaller than a billion.” It’s a false statement, but I can say it.

But just because I can say a statement doesn’t mean that I can prove it true or false. Most of the time, the statement is just very difficult to prove, and so you don’t know how to do it. But it’s also possible to have statements which are impossible to prove either true or false. We’ll call these kind of statements unprovable. Any logical system (set of axioms) with unprovable statements is called incomplete.

Gödel’s first incompleteness theorem says that if you have a consistent mathematical system (i.e., a set of axioms with no contradictions) in which you can do a certain amount of arithmetic, then there are statements in that system which are unprovable using only that system’s axioms.

In other words, math is incomplete. It is impossible to prove everything.

The most basic idea of the proof of the first incompleteness theorem is to think about the statement, “This statement is unprovable.”

If you could prove this statement true, it is by definition provable. But the statement itself says that it is unprovable, and so, since it is true, the statement is also unprovable! But it can’t be both provable and unprovable. Thus the statement must be only unprovable.

While this is the basic idea we’ll employ, the problem is that there isn’t an obvious formal way to say “This statement is unprovable” inside of math. What do you mean by provable? What does “this statement” refer to? Using which axioms?

Gödel’s proof has to make all of that perfectly precise.

The first step is to show that any precise mathematical statement can be transformed into a number, and vice versa.

This step is clever, but not particularly complicated. At some point, you’ve probably come across a code where each letter is exchanged for a number. If we do a to 1, b to 2, etc., for instance, the word “math” would be “13-1-20-8.” Computers use a similar scheme to store text as 1’s and 0’s.

The number Gödel assigns to a precise mathematical statement uses a similar encoding. There are several ways to do this, but I’ll mention a way similar to how Gödel originally did it.

First, associate each mathematical symbol (in your particular mathematical system) with a unique counting number. For instance, maybe “0” is saved as 1, while “=” is saved as 2 and “+” is saved as 3.

An mathematical statement is just a list of these symbols. Equivalently, the statement is a list of the numbers we used to encode the individual symbols. For instance, is equivalent to .

To encode the statement as a single number, we set the Gödel number equal to the the product of the first few primes, raised to the powers in the corresponding position in the list. Thus, the Gödel number of is .

For a statement like “ ”, we’ll use the notation to refer to the Gödel number of that statement. Thus, .

As you can imagine, Gödel numbers can get very large, very quickly, for even moderately long statements. But size is not an issue — we don’t need to write down those numbers, just know they exist.

The key issue is that we can take a number and go backwards to get a mathematical statement.

Every number can be broken up into primes in a unique way. So, , and so the number 145530 represents the statement .

Any precise mathematical statement can be translated into a number this way. Even a proof is just a bunch of statements strung together. (“A” implies “B,” and “B” implies “C,” so “A” implies “C”.) That means we’ve shown that all of math can be written in terms of just numbers.

Similarly, there is a arithmetical way of checking whether a string of statements (as represented by a Gödel number) is a proof of another statement (as represented by another Gödel number.)

While translating any mathematical statement into a number seems like an interesting trick, it turns out to be the key to the proof.

The reason it is so important is that it lets us turn any questions about proofs and provability into an arithmetic question about numbers. Thus, we can use only numbers and their properties in order to prove any (provable) statement.

For instance, consider the statement, which I’ll call . The statement is “ is the Gödel number of a statement, and there does not exist a number which is the Gödel number of a proof of that statement.”

Thus, essentially says “The statement represented by ” is unprovable. But, instead of a question about proofs and statements, it is a statement entirely about numbers, and some arithmetical relationship between them!

The exact arithmetical relationship is very, very complicated, but it can be precisely defined. An analogous, but much simpler statement to could be which we’ll call the statement “ is a prime number.” Thus makes a claim about a number, but that claim can be entirely decided just by some (relatively) simple arithmetic.

We’re coming into the homestretch now.

The original idea for the proof was the statement “This statement is unprovable.” With the precise mathematical statement $\latex Unprovable(y)$, we can make that imprecise statement perfectly precise.

To come up with a precise version of “This statement is unprovable,” we’ll use the “diagonal lemma.” (A lemma is just a theorem you use to prove another theorem.) The diagonal lemma shows that, in the kind of mathematical system we’re using for this proof, there is some statement which is true if and only if is true . (Remember, the input to is a number representing the Gödel number of a statement. In this case, that statement is .)

To be clear, the lemma doesn’t prove that either or is true, only that they are either both true or both false. But what does this mean?

Again, the diagonal lemma shows that (some unknown mathematical statement, probably quite long) is true if and only if is true. But being true means that is unprovable. (That was the definition of .)

So, if we were able to prove the statement true, then the diagonal lemma shows that we can prove true. But says that is unprovable! Thus is both provable and unprovable, a contradiction.

Thus must actually be unprovable.

The statement is the precise version of the statement “This statement is unprovable.” that we were looking for. Thus, not every statement can be proved.

Poor, broken math…

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