As you may be figuring out, there’s a reason why I resisted walking through Gödel’s proof of incompleteness for so long. Incompeteness isn’t a simple proof!

To refresh your memory, here’s a sketch of the proof:

1. Take a simple logic. We’ve been using a variant of the Principia Mathematica’s logic, because that’s what Gödel used.
2. Show that any statement in the logic can be encoded as a number using an arithmetic process based on the syntax of the logic. The process of encoding statements numerically is called Gödel numbering.
3. Show that you can express meta-mathematical properties of logical statements in terms of arithemetic properties of their Gödel numbers. In particular, we need to build up the logical infrastructure that we need to talk about whether or not a statement is provable.
4. Using meta-mathematical properties, show how you can create an unprovable statement encoded as a Gödel number.

What we’ve done so far is the first two steps, and part of the third. In this post, we saw the form of the Principia’s logic that we’re using, and how to numerically encode it as a Gödel numbering. We’ve start started on the third point in this post, by figuring out just what it means to say that things are encoded arithmetically. Now we can get to the part where we see how to encode meta-mathematical properties in terms of arithmetic properties of the Gödel numbering. In this post, we’re going to build up everything we need to express syntactic correctness, logical validity, and provability in terms of arithmetical properties of Gödel numbers. (And, as a reminder, I’ve been using this translation on Gödel’s original paper on incompleteness.)

This is the most complex part of the incompleteness proof. The basic concept of what we’re doing is simple, but the mechanics are very difficult. What we want to do is define a set of predicates about logical statements, but we want those predicates to be expressed as arithmetic properties of the numerical representations of the logical statements.

The point of this is that we’re showing that done in the right way, arithmetic is logic – that doing arithmetic on the Gödel numbers is doing logical inference. So what we need to do is build up a toolkit that shows us how to understand and manipulate logic-as-numbers using arithmetic. As we saw in the last post, primitive recursion is equivalent to arithmetic – so if we can show how all of the properties/predicates that we define are primitive recursive, then they’re arithmetic.

This process involves a lot of steps, each of which is building the platform for the steps that follow it. I struggled quite a bit figuring out how to present these things in a comprehensible way. What I ended up with is writing them out as code in a pseudo-computer language. Before inventing this language, I tried writing actual executable code, first in Python and then in Haskell, but I wasn’t happy with the clarity of either one.

Doing it in an unimplemented language isn’t as big a problem as you might think. Even if this was all executable, you’re not going to be able to actually run any of it on anything real – at least not before you hair turns good and gray. The way that this stuff is put together is not what any sane person would call efficient. But the point isn’t to be efficient: it’s to show that this is possible. This code is really all about searching; if we wanted to be efficient, this could all be done in a different representation, with a different search method that was a lot faster – but that wolud be harder to understand.

So, in the end, I threw together a simple language that’s easy to read. This language, if it were implemented, wouldn’t really even be Turing complete – it’s a primitive recursive language.

Basics

We’ll start off with simple numeric properties that have no obvious connection to the kinds of meta-mathematical statements that we want to talk about, but we’ll use those to define progressively more and more complex and profound properties, until we finally get to our goal.

# divides n x == True if n divides x without remainder.
pred divides(n, x) = x mod n == 0

pred isPrime(0) = False
pred isPrime(1) = False
pred isPrime(2) = True
pred isPrime(n) = {
  all i in 2 to n {
    not divides(i, n)
  }
}

fun fact(0) = 1
fun fact(n) = n * fact(n - 1)

Almost everything we’re going to do here is built on a common idiom. For anything we want to do arithmetically, we’re going to find a bound – a maximum numeric value for it. Then we’re going to iterate over all of the values smaller than that bound, searching for our target.

For example, what’s the nth prime factor of x? Obviously, it’s got to be smaller than x, so we’ll use x as our bound. (A better bound would be the square root of x, but it doesn’t matter. We don’t care about efficiency!) To find the nth prime factor, we’ll iterate over all of the numbers smaller than our bound x, and search for the smallest number which is prime, which divides x, and which is larger than the n-1th prime factor of x. We’ll translate that into pseudo-code:

fun prFactor(0, x) = 0
fun prFactor(n, x) = {
  first y in 1 to x {
    isPrime(y) and divides(y, x) and prFactor(n - 1, x) < y
  }
}

Similarly, for extracting values from strings, we need to be able to ask, in general, what's the nth prime number? This is nearly identical to prFactor above. The only difference is that we need a different bound. Fortunately, we know that the nth prime number can't be larger than the factorial of the previous prime plus 1.

fun nthPrime(0) = 0
fun nthPrime(n) = {
  first y in 1 to fact(nthPrime(n - 1)) + 1  {
    isPrime(y) and y > nthPrime(n - 1))
  }
}

In composing strings of Gödel numbers, we use exponentiation. Given integers x and n, xⁿ, we can obviously compute them via primitive recursion. I'll define them below, but in the rest of this post, I'll write them as an operator in the language:

fun pow(n, 0) = 1
fun pow(n, i) = n * pow(n, i - 1)

String Composition and Decomposition

With those preliminaries out of the way, we can get to the point of defining something that's actually about one of the strings encoded in these Gödel numbers. Given a number n encoding a string, item(n, x) is the value of the nth character of x. (This is slow. This is really slow! We're getting to the limit of what a very powerful computer can do in a reasonable amount of time. But this doesn't matter. The point isn't that this is a good way of doing these things: it's that these things are possible. To give you an idea of just how slow this is, I started off writing the stuff in this post in Haskell. Compiled with GHC, which is a very good compiler, item to extract the 6th character of an 8 character string took around 10 minutes on a 2.4Ghz laptop.)

fun item(n, x) = {
  first y in 1 to x {
    divides(prFactor(n, x) ** y, y) and
      not divides(prFactor(n, x)**(y+1), x)
  }
}

Given a string, we want to be able to ask how long it is; and given two strings, we want to be able to concatenate them.

fun length(x) = {
  first y in 1 to x {
    prFactor(y, x) > 0 and prFactor(y + 1, x) == 0
  }
}

fun concat(x, y) = {
  val lx = length(x)
  val ly = length(y)

  first z in 1 to nthprime(lx + ly)**(x + y) {
    (all n in 1 to lx {
        item(n, z) == item(n, x)
     }) and (all n in 1 to ly {
        item(n + lx, z) == item(n, y)
      })
  }
}

fun concatl([]) = 0
fun concatl(xs) = {
  concat(head(xs), concatl(tail(xs)))
}

fun seq(x) = 2**x

We want to be able to build statements represented as numbers from other statements represented as numbers. We'll define a set of functions that either compose new strings from other strings, and to check if a particular string is a particular kind of syntactic element.

# x is a variable of type n.
pred vtype(n, x) = {
  some z in 17 to x {
    isPrime(z) and x == n**z
  }
}

# x is a variable
pred isVar(x) = {
  some n in 1 to x {
    vtype(n, x)
  }
}

fun paren(x) =
  concatl([gseq(11), x, gseq(13)])

# given the Gödel number for a statement x, find
# the Gödel number for not x.
fun gnot(x) =
  concat(gseq(5), paren(x))

# Create the number for x or y.
fun gor(x, y) =
  concatl([paren(x), seq(7), paren(y)])

# Create the number for 'forall x(y)'.
fun gforall(x, y) =
  concatl([seq(9), seq(x), paren(y)])

# Create the number for x with n invocations of the primitive
# successor function.
fun succn(0, x) = x
fun succn(n, x) = concat(seq(3), succn(n - 1, x))

# Create the number n using successor and 0.
fun gnumber(n) = succn(n, seq(1))

# Check if a statement is type-1.
pred stype_one(x) = {
  some m in 1 to x {
     m == 1 or (vtype(1, m) and x == succn(n, seq(m))
  }
}

# Check if a statement is type n.
pred fstype(1, x) = stype_one(x)
pred fstype(n, x) =
  some v in 1 to x {
    vtype(n, v) and R(v)
  }
}

That last function contains an error: the translation of Gödel that I'm using says R(v) without defining R. Either I'm missing something, or the translator made an error.

Formulae

Using what we've defined so far, we're now ready to start defining formulae in the basic Principia logic. Forumlae are strings, but they're strings with a constrained syntax.

pred elFm(x) = {
  some y in 1 to x {
    some z in 1 to x {
      some n in 1 to x {
        stype(n, y) and stype(n+1, z) and x == concat(z, paren(y))
      }
    }
  }
}

All this is doing is expressing the grammar rule in arithmetic form: an elementary formula is a predicate: P(x), where x is a variable on level n, and P is a variable of level x + 1.

The next grammar rule that we encode this way says how we can combine elementary formulae using operators. There are three operators: negation, conjunction, and universal quantification.

pred op(x, y, z) = {
  x == gnot(y) or
  x == gor(y, z) or
  (some v in 1 to x { isVar(v) and x == gforall(v, y) })
}

And now we can start getting complex. We're going to define the idea of a valid sequence of formulae. x is a valid sequence of formulae when it's formed from a collection of formulae, each of which is either an elementary formula, or is produced from the formulae which occured before it in the sequence using either negation, logical-or, or universal quantification.

In terms of a more modern way of talking about it, the syntax of the logic is a grammar. A formula sequence, in this system, is another way of writing the parse-tree of a statement: the sequence is the parse-tree of the last statement in the sequence.

pred fmSeq(x) = {
  all p in 0 to length(x) {
    elFm(item(n, x)) or
      some p in 0 to (n - 1) {
        some q in 0 to (n - 1) {
          op(item(n,x), item(p, x), item(q, x))
        }
      }
  }
}

The next one bugs me, because it seems wrong, but it isn't really! It's a way of encoding the fact that a formula is the result of a well-defined sequence of formulae. In order to ensure that we're doing primitive recursive formulae, we're always thinking about sequences of formulae, where the later formulae are produced from the earlier ones. The goal of the sequence of formula is to produce the last formula in the sequence. What this predicate is really saying is that a formula is a valid formula if there is some sequence of formulae where this is the last one in the sequence.

Rephrasing that in grammatical terms, a string is a formula if there is valid parse tree for the grammar that produces the string.

pred isFm(x) = {
  some n in 1 to nthPrime(length(x)**2)**(x*length(x)**2) {
    fmSeq(n)
  }
}

So, now, can we say that a statement is valid because it's parsed according to the grammar? Not quite. It's actually a familiar problem for people who write compilers. When you parse a program in some language, the grammar doesn't usually specify variables must be declared before they're used. It's too hard to get that into the grammar. In this logic, we've got almost the same problem: the grammar hasn't restricted us to only use bound variables. So we need to have ways to check whether a variable is bound in a Gödel-encoded formula, and then use that to check the validity of the formula.

# The variable v is bound in formula x at position n.
pred bound(v, n, x) = {
  isVar(v) and isFm(x) and
  (some a in 1 to x {
    some b in 1 to x {
      some c in 1 to x {
        x == concatl([a, gforall(v, b), c]) and
        isFm(b) and
        length(a) + 1 ≤ n ≤ length(a) + length(forall(v, b))
      }
    }
  })
}

# The variable v in free in formula x at position n
pred free(v, n, x) = {
  isVar(v) and isFm(x) and
  (some a in 1 to x {
    some b in 1 to x {
      some c in 1 to x {
        v == item(n, x) and n ≤ length(x) and not bound(v, n, x)
      }
    }
  })
}

pred free(v, x) = {
  some n in 1 to length(x) {
    free(v, n, x)
  }
}

To do logical inference, we need to be able to do things like replace a variable with a specific infered value. We'll define how to do that:

# replace the item at position n in x with y.
fun insert(x, n, y) = {
  first z in 1 to nthPrime(length(x) + length(y))**(x+y) {
    some u in 1 to x {
      some v in 1 to x {
        x == concatl([u, seq(item(n, x)), v]) and
        z == concatl([u, y, v]) and
        n == length(u) + 1
      }
    }
  }
}

There are inference operations and validity checks that we can only do if we know whether a particular variable is free at a particular position.

# freePlace(k, v, k) is the k+1st place in x (counting from the end)
# where v is free.
fun freePlace(0, v, x) = {
  first n in 1 to length(x) {
    free(v, n, x) and
    not some p in n to length(x) {
      free(v, p, x)
    }
  }
}

fun freePlace(k, v, x) = {
  first n in 1 to freePlace(n, k - 1, v) {
    free(v, n, x) and
    not some p in n to freePlace(n, k - 1, v) {
      free(v, p, x)
    }
  }
}

# number of places where v is free in x
fun nFreePlaces(v, x) = {
  first n in 1 to length(x) {
    freeplace(n, v, x) == 0
  }
}

In the original logic, some inference rules are defined in terms of a primitive substitution operator, which we wrote as subst[v/c](a) to mean substitute the value c for the variable c in the statement a. We'll build that up on a couple of steps, using the freePlaces function that we just defined.

# Subst1 replaces a single instance of v with y.
fun subst'(0, x, v, y) = x
fun subst1(0k, x, v, y) =
  insert(subst1(k, x, v, y), freePlace(k, v, x), y)

# subst replaces all instances of v with y
fun subst(x, v, y) = subst'(nFreePlaces(v, x), x, v, y)

The next thing we're going to do isn't, strictly speaking, absolutely necessary. Some of the harder stuff we want to do will be easier to write using things like implication, which aren't built in primitive of the Principia logic. To write those as clearly as possible, we'll define the full suite of usual logical operators in terms of the primitives.

# implication
fun gimp(x, y) = gor(gnot(x), y)

# logical and
fun gand(x, y) = gnot(gor(gnot(x), gnot(y)))

# if/f
fun gequiv(x, y) = gand(gimp(x, y), gimp(y, x))

# existential quantification
fun gexists(v, y) = not(gforall(v, not(y)))

Axioms

The Peano axioms are valid logical statements, so they have Gödel numbers in this system. We could compute their value, but why bother? We know that they exist, so we'll just give them names, and define a predicate to check if a value matches them.

The form of the Peano axioms used in incompleteness are:

1. Zero: ¬(succ(x₁) = 0)
2. Uniqueness: succ(x₁) = succ(y₁) Rightarrow x = y
3. Induction: x₂(0) ∧ ∀x₁(x₂(x₁)⇒ x₂(succ(x₁))) ⇒ ∀x₁(x₂(x₁))

const pa1 = ...
const pa2 = ...
const pa3 = ...

pred peanoAxiom(x) =
  (x == pa1) or (x == pa2) or (x == pa3)

Similarly, we know that the propositional axioms must have numbers. The propositional
axioms are:

I'll show the translation of the first - the rest follow the same pattern.

# Check if x is a statement that is a form of propositional
# axiom 1: y or y => y
pred prop1Axiom(x) =
  some y in 1 to x {
    isFm(x) and x == imp(or(y, y), y)
  }
}

pred prop2Axiom(x) = ...
pred prop3Axiom(x) = ...
pred prop4Axiom(x) = ...
pred propAxiom(x) = prop2Axiom(x) or prop2Axiom(x) or
    prop3Axiom(x) or prop4Axiom(x)

Similarly, all of the other axioms are written out in the same way, and we add a predicate isAxiom to check if something is an axiom. Next is quantifier axioms, which are complicated, so I'll only write out one of them - the other follows the same basic scheme.

The two quantifier axioms are:

quantifier_axiom1_condition(z, y, v) = {
  not some n in 1 to length(y) {
    some m in 1 to length(z) {
      some w in 1 to z {
         w == item(m, z) and bound(w, n, y) and free(v, n, y)
      }
    }
  }
}

pred quantifier1Axiom(x) = {
  some v in 1 to x {
    some y in 1 to x {
      some z in 1 to x {
        some n in 1 to x {
          vtype(n, v) and stype(n, z) and
          isFm(y) and
          quantifier_axiom1_condition(z, y, v) and
          x = gimp(gforall(v, y), subst(y, v, z))
        }
      }
    }
  }
}

quanitifier_axiom2 = ...
isQuantifierAxiom = quantifier1Axiom(x) or quantifier2Axiom(x)

We need to define a predicate for the reducibility axiom (basically, the Principia's version of the ZFC axiom of comprehension). The reducibility axiom is a schema: for any predicate , . In our primitive recursive system, we can check if something is an instance of the reducibility axiom schema with:

pred reduAxiom(x) =
  some u in 1 to x {
    some v in 1 to x {
      some y in 1 to x {
        some n in 1 to x {
          vtype(n, v) and
          vtype(n+1, u) and
          not free(u, y) and
          isFm(y) and
          x = gexists(u, gforall(v, gequiv(concat(seq(u), paren(seq(v))), y)))
        }
      }
    }
  }
}

Now, the set axiom. In the logic we're using, this is the axiom that defines set equality. It's written as . Set equality is defined for all types of sets, so we need to have one version of axiom for each level. We do that using type-lifting: we say that the axiom is true for type-1 sets, and that any type-lift of the level-1 set axiom is also a version of the set axiom.

fun typeLift(n, x) = {
  first y in 1 to x**(x**n) {
    all k in 1 to length(x) {
      item(k, x) ≤ 13 and item(k, y) == item(k, v) or
      item(k, x) > 13 and item(k, y) = item(k, x) * prFactor(1, item(k, x))**n
    }
  }
}

We haven't defined the type-1 set axiom. But we just saw the axiom above, and it's obviously a simple logical statement. That mean that it's got a Gödel number. Instead of computing it, we'll just say that that number is called sa1. Now we can define a predicate to check if something is a set axiom:

val sa1 = ...
pred setAxiom(x) =
  some n in 1 to x {
    x = typeLift(n, sa)
  }
}

We've now defined all of the axioms of the logic, so we can now create a general predicate to see if a statement fits into any of the axiom categories:

pred isAxiom(x) =
  peanoAxiom(x) or propAxiom(x) or quantifierAxom(x) or
  reduAxiom(x) or setAxiom(x)

Proofs and Provability!

With all of the axioms expressible in primitive recursive terms, we can start on what it means for something to be provable. First, we'll define what it means for some statement x to be an immediate consequence of some statements y and z. (Back when we talked about the Principia's logic, we said that x is an immediate consequence of y and z if either: y is the formula z ⇒ x, or if c is the formula ∀v.x).

pred immConseq(x, y, z) = {
  y = imp(z, x) or
  some v in 1 to x {
    isVar(v) and x = forall(v, y)
  }
}

Now, we can use our definition of an immediate consequence to specify when a sequence of formula is a proof figure. A proof figure is a sequence of statements where each statement in it is either an axiom, or an immediate consequence of two of the statements that preceeded it.

pred isProofFigure(x) = {
  (all n in 0 to length(x) {
    isAxiom(item(n, x)) or
    some p in 0 to n {
      some q in 0 to n {
        immConseq(item(n, x), item(p, x), item(q, x))
      }
    }
  }) and
  length(x) > 0
}

We can say that x is a proof of y if x is proof figure, and the last statement in x is y.

pred proofFor(x, y) =
  isProofFigure(x) and
  item(length(x), x) == y

Finally, we can get to the most important thing! We can define what it means for something to be provable! It's provable if there's a proof for it!

pre provable(x) =
  some y {
    proofFor(y, x)
  }
}

Note that this last one is not primitive recursive! There's no way that we can create a bound for this: a proof can be any length.

At last, we're done with these definition. What we've done here is really amazing: now, every logical statement can be encoded as a number. Every proof in the logic can be encoded as a sequence of numbers: if something is provable in the Principia logic, we can encode that proof as a string of numbers, and check the proof for correctness using nothing but (a whole heck of a lot of) arithmetic!

Next post, we'll finally get to the most important part of what Gödel did. We've been able to define what it means for a statement to be provable - we'll use that to show that there's a way of creating a number encoding the statement that something is not provable. And we'll show how that means that there is a true statement in the Principia's logic which isn't provable using the Principia's logic, which means that the logic isn't complete.

In fact, the proof that we'll do shows a bit more than that. It doesn't just show that the Principia's logic is incomplete. It shows that any consistent formal system like the Principia, any system which is powerful enough to encode Peano arithmetic, must be incomplete.

When I left off, we’d seen how to take statements written in the logic of the Principia Mathematica, and convert them into numerical form. What we need to see now is how to get meta-mathematical.

We want to be able to write second-order logical statements. The basic trick to incompleteness is that we’re going to use the numerical encoding of statements to say that a predicate or relation is represented by a number. Then we’re going to write predicates about predicates by defining predicates on the numerical representations of the first-order predicates. That’s going to let us create a true statement in the logic that can’t be proven with the logic.

To do that, we need to figure out how to take our statements and relations represented as numbers, and express properties of those statements and relations in terms of arithmetic. To do that, we need to define just what it means to express something arithmetically. Gödel did that by defining “arithmetically” in terms of a concept called primitive recursion.

I learned about primitive recursion when I studied computational complexity. Nowadays, it’s seen as part of theoretical computer science. The idea, as we express it in modern terms, is that there are many different classes of computable functions. Primitive recursion is one of the basic complexity classes. You don’t need a Turing machine to compute primitive recursive functions – they’re a simpler class.

The easiest way to understand primitive recursion is that it’s what you get in a programming language with integer arithmetic, and simple for-loops. The only way you can iterate is by repeating things a bounded number of times. Primitive recursion has a lot of interesting properties: the two key ones for our purposes here are: number theoretic proofs are primitive recursive, and every computation of a primitive recursive function is guaranteed to complete within a bounded amount of time.

The formal definition of primitive recursion, the way that Gödel wrote it, is quite a bit more complex than that. But it means the same thing.

We start with what it means to define a formula via primitive recursion. (Note the language that I used there: I’m not explaining what it means for a function to be primitive recursive; I’m explaining what it means to be defined via primitive recursion.) And I’m defining formulae, not functions. In Gödel’s proof, we’re always focused on numerical reasoning, so we’re not going to talk about programs or algorithms, we’re going to about the definition of formulae.

A formula is defined via primitive recursion if, for some other formulae and :

• Base:
• Recursive: .

So, basically, the first parameter is a bound on the number of times that can invoked recursively. When it’s 0, you can’t invoke any more.

A formula is primitive recursive if it defined from a collection of formulae where any formula is defined via primitive recursion from , or the primitive succ function from Peano arithmetic.

For any formula in that sequence, the degree of the formula is the number of other primitive recursive formulae used in its definition.

Now, we can define a primitive recursive property: is primitive recursive if and only if there exists a primitive recursive function such that .

With primitive recursive formulae and relations defined, there’s a bunch of theorems about how you can compose primitive recursive formulae and relations:

1. Every function or relation that you get by substituting a primitive recursive function for a variable in a primitive recursive function/relation is primitive recursive.
2. If R and S are primitive relations, then ¬R, R∧S, R∨S are all primitive recursive.
3. If and are primitive recursive functions, then the relation is also primitive recursive.
4. Let and be finite-length tuples of variables. If the function and the relation are primitive recursive, then so are the relations:
   • 
   • 
5. Let and be finite-length tuples of variables. And let be the smallest value of for which and is true, or 0 if there is no such value. Then if the function and the relation are primitive recursive, then so is the function .

By these definitions, addition, subtraction, multiplication, and integer division are all primitive recursive.

Ok. So, now we’ve got all of that out of the way. It’s painful, but it’s important. What we’ve done is come up with a good formal description of what it means for something to be an arithmetic property: if we can write it as a primitive recursive relation or formula, it’s arithmetic.

So now, finally, we’re ready to get to the really good part! Now that we know what it means to define something arithmetically, we can see how to define meta-mathematical properties and formulae arithmetically. Next post, hopefully tomorrow, I’ll start showing you arithmetic expressions of meta-math!

The first step in Gödel’s incompleteness proof was finding a way of taking logical statements and encoding them numerically. Looking at this today, it seems sort-of obvious. I mean, I’m writing this stuff down in a text file – that text file is a stream of numbers, and it’s trivial to convert that stream of numbers into a single number. But when Gödel was doing it, it wasn’t so obvious. So he created a really clever mechanism for numerical encoding. The advantage of Gödel’s encoding is that it makes it much easier to express properties of the encoded statements numerically.

Before we can look at how Gödel encoded his logic into numbers, we need to look at the logic that he used. Gödel worked with the specific logic variant used by the Principia Mathematica. The Principia logic is minimal and a bit cryptic, but it was built for a specific purpose: to have a minimal syntax, and a complete but minimal set of axioms.

The whole idea of the Principia logic is to be purely syntactic. The logic is expected to have a valid model, but you shouldn’t need to know anything about the model to use the logic. Russell and Whitehead were deliberately building a pure logic where you didn’t need to know what anything meant to use it. I’d really like to use Gödel’s exact syntax – I think it’s an interestingly different way of writing logic – but I’m working from a translation, and the translator updated the syntax. I’m afraid that switching between the older Gödel syntax, and the more modern syntax from the translation would just lead to errors and confusion. So I’m going to stick with the translation’s modernization of the syntax.

The basic building blocks of the logic are variables. Already this is a bit different from what you’re probably used to in a logic. When we think of logic, we usually consider predicates to be a fundamental thing. In this logic, they’re not. A predicate is just a shorthand for a set, and a set is represented by a variable.

Variables are stratified. Again, it helps to remember where Russell and Whitehead were coming from when they were writing the Principia. One of their basic motivations was avoiding self-referential statements like Russell’s paradox. In order to prevent that, they thought that they could create a stratified logic, where on each level, you could only reason about objects from the level below. A first-order predicate would be a second-level object could only reason about first level objects. A second-order predicate would be a third-level object which could reason about second-level objects. No predicate could ever reason about itself or anything on its on level. This leveling property is a fundamental property built into their logic. The way the levels work is:

• Type one variables, which range over simple atomic values, like specific single natural numbers. Type-1 variables are written as a₁, b₁.
• Type two variables, which range over sets of atomic values, like sets of natural numbers. A predicate, like IsOdd, about specific natural numbers would be represented as a type-2 variable. Type-2 variables are written a₂, b₂, …
• Type three variables range over sets of sets of atomic values. The mappings of a function could be represented as type-3 variables: in set theoretic terms, a function is set of ordered pairs. Ordered pairs, in turn, can be represented as sets of sets – for example, the ordered pair (1, 4) would be represented by the set { {1}, {1, 4} }. A function, in turn, would be represented by a type-4 variable – a set of ordered pairs, which is a set of sets of sets of values.

Using variables, we can form simple atomic expressions, which in Gödel’s terminology are called signs. As with variables, the signs are divided into stratified levels:

• Type-1 signs are variables, and successor expressions. Successor expressions are just Peano numbers written with “succ”: 0, succ(0), succ(succ(0)), succ(a₁), etc.
• Signs of any type greater than 1 are just variables of that type/level.

Now we can assemble the basic signs into formulae. Gödel explained how to build formulae in a classic logicians form, which I think is hard to follow, so I’ve converted it into a grammar:

 elementary_formula → signn+1(signn)
 formula → ¬(elementary_formula)
 formula → (elementary_formula) or (elementary_formula)
 formula → ∀ signn (elementary_formula)

That’s all that’s really included in Gödel’s logic. It’s enough: everything else can be defined in terms of combinatinos of these. For example, you can define logical and in terms of negation and logical or: (a ∧ b) ⇔ ¬ (¬ a ∨ ¬ b).

Next, we need to define the axioms of the system. In terms of logic the way I think of it, these axioms include both “true” axioms, and the inference rules defining how the logic works. There are five families of axioms.

<

ol>

<ol>
   <li> <em>p &or; p &rArr; p</em></li>
   <li> <em>p &rArr; p &or; q</em></li>
   <li> <em>p &or; q &rArr; q &or; p</em></li>
   <li> <em>(p &rArr; q) &rArr; (p &or; r) &rArr; (q &or; r)</em></li>

<li> And last but not least, an axiom defining set equivalence:       <em>&forall; x<sub>i</sub> (x<sub>i+1</sub>(x<sub>i</sub>) &hArr; y<sub>i+1</sub>(y<sub>i</sub>)) &rArr; x<sub>i+1</sub> = y<sub>i+1</sub></em>   </li>

So, now, finally, we can get to the numbering. This is quite clever. We’re going to use the simplest encoding: for every possible string of symbols in the logic, we’re going to define a representation as a number. So in this representation, we are not going to get the property that every natural number is a valid formula: lots of natural numbers won’t be. They’ll be strings of nonsense symbols. (If we wanted to, we could make every number be a valid formula, by using a parse-tree based numbering, but it’s much easier to just let the numbers be strings of symbols, and then define a predicate over the numbers to identify the ones that are valid formulae.)

We start off by assigning numbers to the constant symbols:

Variables will be represented by powers of prime numbers, for prime numbers greater that 13. For a prime number p, p will represent a type one variable, p² will represent a type two variable, p³ will represent a type-3 variable, etc.

Using those symbol encodings, we can take a formula written as symbols x₁x₂x₃…x_n, and encode it numerically as the product 2^x₁3^x₂5^x₂…p_n^x_n, where p_n is the nth prime number.

For example, suppose I wanted to encode the formula: ∀ x₁ (y₂(x₁)) ∨ x₂(x₁).

First, I’d need to encode each symbol:

1. “∀” would be 9.
2. “x₁“” = 17
3. “(” = 11
4. “y₂” = 19² = 361
5. “(” = 11
6. “x₁” = 17
7. “)” = 13
8. “∨” = 7
9. “x₂” = 17² = 289
10. “(” = 11
11. “x₁” = 17
12. “)” = 13
13. “)” = 13

The formula would thus be turned into the sequence: [9, 17, 11, 361, 11, 17, 13, 7, 289, 11, 17, 13, 13]. That sequence would then get turned into a single number 2⁹ 3¹⁷ 5¹¹ 7³⁶¹ 11¹¹ 13¹⁷ 17¹³ 19⁷ 23²⁸⁹ 29¹¹ 31¹⁷ 37¹³ 41¹³, which according to Hugs is the number (warning: you need to scroll to see it. a lot!):

1,821,987,637,902,623,701,225,904,240,019,813,969,080,617,900,348,538,321,073,935,587,788,506,071,830,879,280,904,480,021,357,988,798,547,730,537,019,170,876,649,747,729,076,171,560,080,529,593,160,658,600,674,198,729,426,618,685,737,248,773,404,008,081,519,218,775,692,945,684,706,455,129,294,628,924,575,925,909,585,830,321,614,047,772,585,327,805,405,377,809,182,961,310,697,978,238,941,231,290,173,227,390,750,547,696,657,645,077,277,386,815,869,624,389,931,352,799,230,949,892,054,634,638,136,137,995,254,523,486,502,753,268,687,845,320,296,600,343,871,556,546,425,114,643,587,820,633,133,232,999,109,544,763,520,253,057,252,248,982,267,078,202,089,525,667,161,691,850,572,084,153,306,622,226,987,931,223,193,578,450,852,038,578,983,945,920,534,096,055,419,823,281,786,399,855,459,394,948,921,598,228,615,703,317,657,117,593,084,977,371,635,801,831,244,944,526,230,994,115,900,720,026,901,352,169,637,434,441,791,307,175,579,916,097,240,141,893,510,281,613,711,253,660,054,258,685,889,469,896,461,087,297,563,191,813,037,946,176,250,108,137,458,848,099,487,488,503,799,293,003,562,875,320,575,790,915,778,093,569,309,011,025,000,000,000.

Next, we’re going to look at how you can express interesting mathematical properties in terms of numbers. Gödel used a property called primitive recursion as an example, so we’ll walk through a definition of primitive recursion, and show how Gödel expressed primitive recursion numerically.