Gödel Numbering in Go: A Mathematical Quiche

GoQuiche transforms Go source code into a single, mathematically significant integer through Gödel numbering. While seemingly absurd at first glance, this project explores the intersection of programming language theory and mathematical logic, demonstrating how any formal system can be encoded uniquely into arithmetic. Though the resulting numbers are impractically large, this technique forms the foundation of Gödel’s incompleteness theorems and provides insight into theoretical computer science’s relationship with formal systems.

1. Introduction

Imagine taking an entire program, with all its functions, variables, and logic, and representing it as a single number. Not a binary number or a hexadecimal value as we normally think of code, but a single, admittedly massive, integer that would make even mathematicians question your life choices. This is essentially what Gödel numbering achieves, and it’s the principle behind GoQuiche.

I created GoQuiche partly as a mathematical curiosity, partly as a demonstration of Gödel’s encoding technique, and partly because the idea of baking programs into “quiches” made of prime numbers amused me. The name itself is a play on Gödel’s technique and the culinary metaphor of taking separate ingredients and combining them into a unified dish – though unlike real quiche, this one is guaranteed to be indigestible to both humans and computers alike.

2. The Mathematical Foundation

Kurt Gödel, in proving his famous incompleteness theorems, developed a technique to encode formal mathematical statements as unique natural numbers. He did this by assigning prime numbers to symbols and using exponentiation to create a unique mapping for any sequence of symbols.

The process is elegantly simple in concept:

1. Assign a unique prime number to each symbol in your “alphabet”
2. For a sequence of symbols, raise each prime to a power corresponding to its position
3. Multiply these values together to get a unique number

This creates a one-to-one mapping between sequences and numbers, allowing mathematical operations to stand in for syntactic operations.

For a simple example, if:
'a' = 2, 'b' = 3, 'c' = 5

Then "abc" would be:
2^1 × 3^2 × 5^3 = 2 × 9 × 125 = 2250

This process is reversible (though computationally intensive), meaning you can recover the original sequence from just the number.

3. GoQuiche Implementation

GoQuiche applies this concept to Go source code. It tokenizes the input program, assigns prime numbers to each unique token, and then computes the corresponding Gödel number.

The core of the implementation revolves around these key steps:

1. Lexical analysis of Go source code using Go’s own scanner package
2. Generation of prime numbers as needed
3. Assignment of primes to each unique symbol
4. Computation of the final Gödel number via exponentiation
5. A reversal function to demonstrate the recoverability

Let’s break down the process with the actual implementation.

3.1. Prime Number Generation

First, we need a source of prime numbers:

func generatePrimes(n int) []int {
	primes := []int{2}
	num := 3
	for len(primes) < n {
		isPrime := true
		for _, prime := range primes {
			if num%prime == 0 {
				isPrime = false
				break
			}
			if prime*prime > num {
				break
			}
		}
		if isPrime {
			primes = append(primes, num)
		}
		num += 2
	}
	return primes
}

This function implements a straightforward sieve method to generate the first n prime numbers, which will be assigned to our tokens.

3.2. Tokenization and Prime Assignment

The next step uses Go’s scanner package to break down the source code into tokens:

// Initialize the scanner
var s scanner.Scanner
fset := token.NewFileSet()
file := fset.AddFile(os.Args[1], fset.Base(), len(src))
s.Init(file, src, nil, scanner.ScanComments)

// Map to store symbol-prime associations
symbolPrimes := make(map[string]int)
symbolOrder := make([]string, 0)

// Perform lexical analysis
for {
	pos, tok, lit := s.Scan()
	if tok == token.EOF {
		break
	}

	symbol := tok.String()
	if lit != "" {
		symbol = lit
	}

	// Replace newline characters with \n
	symbol = strings.ReplaceAll(symbol, "\n", "\\n")

	if _, exists := symbolPrimes[symbol]; !exists {
		symbolPrimes[symbol] = primes[len(symbolOrder)]
		symbolOrder = append(symbolOrder, symbol)
	}

	fmt.Printf("%s\t%s\t%d\n", fset.Position(pos), symbol, symbolPrimes[symbol])
}

Each unique token is assigned the next available prime number, and we keep track of both the assignments and the order in which symbols appear.

3.3. Computing the Gödel Number

With our symbols assigned to primes, we now compute the Gödel number:

// Print the number line with exponentiated values
fmt.Println("\nNumber line with exponentiated values:")
result := big.NewInt(1)
for i, symbol := range symbolOrder {
	if i > 0 {
		fmt.Print(" * ")
	}
	prime := big.NewInt(int64(symbolPrimes[symbol]))
	exponent := big.NewInt(int64(i + 1))
	value := new(big.Int).Exp(prime, exponent, nil)
	fmt.Printf("%s^%d", symbol, i+1)
	result.Mul(result, value)
}
fmt.Println()
fmt.Printf("\nFinal result: %s\n", result.String())

We use Go’s big.Int for the calculations since the numbers get astronomically large very quickly.

3.4. Recovering the Original Program

To demonstrate that this process is reversible, GoQuiche includes a function to recover the original tokens from the Gödel number:

func recoverSymbols(result *big.Int, symbolPrimes map[string]int) []string {
	// Create a reverse mapping of primes to symbols
	primeToSymbol := make(map[int]string)
	for symbol, prime := range symbolPrimes {
		primeToSymbol[prime] = symbol
	}

	// Sort primes in descending order
	primes := make([]int, 0, len(symbolPrimes))
	for _, prime := range symbolPrimes {
		primes = append(primes, prime)
	}
	sort.Sort(sort.Reverse(sort.IntSlice(primes)))

	recoveredSymbols := []string{}
	n := new(big.Int).Set(result)
	zero := big.NewInt(0)

	for n.Cmp(big.NewInt(1)) > 0 {
		for _, prime := range primes {
			primeBig := big.NewInt(int64(prime))
			exponent := 0
			for new(big.Int).Mod(n, primeBig).Cmp(zero) == 0 {
				n.Div(n, primeBig)
				exponent++
			}
			if exponent > 0 {
				symbol := primeToSymbol[prime]
				recoveredSymbols = append(recoveredSymbols, symbol)
				break
			}
		}
	}

	// Reverse the recovered symbols to get the correct order
	for i := 0; i < len(recoveredSymbols)/2; i++ {
		j := len(recoveredSymbols) - 1 - i
		recoveredSymbols[i], recoveredSymbols[j] = recoveredSymbols[j], recoveredSymbols[i]
	}

	return recoveredSymbols
}

The recovery process divides the Gödel number by prime numbers in descending order, rebuilding the sequence of tokens from their encoded form.

4. Practical Implications and Limitations

While GoQuiche is a fascinating demonstration of Gödel numbering, it quickly runs into practical limitations:

4.1. Astronomical Numbers

Even for small programs, the resulting Gödel numbers become astronomically large. Consider that for a typical “Hello World” program in Go, which might have around 20-30 tokens, we end up with a number that has hundreds of digits. For any real-world program, these numbers would have more digits than there are atoms in the observable universe. Trying to print the Gödel number for even a modest application would likely result in painful consequences to your CPU temps.

4.2. Recovery Complexity

While recovery is theoretically straightforward, the actual process becomes computationally challenging as the numbers grow. The time complexity increases dramatically with program size. But hey, security through obscurity can be argued to work if your threat model supposes that no one wants to bother decoding your giant Gödel number!

That being said, these limitations make GoQuiche impractical for any actual code representation or compression ¹ .

5. Theoretical Significance

Despite its practical limitations, the concept demonstrated by GoQuiche has profound theoretical implications:

5.1. Gödel’s Incompleteness Theorems

Gödel used this numbering technique to prove that any formal system capable of expressing basic arithmetic must contain statements that cannot be proved or disproved within the system ^[1]. By encoding syntactic properties into numbers, he showed that self-reference can occur within formal systems, leading to paradoxes similar to “This statement is unprovable.”

5.2. Computability Theory

The technique demonstrates an important principle in theoretical computer science: programming languages, formal systems, and computational processes can all be encoded as pure mathematics. This connection forms the basis for much of computability theory, from Turing machines to the halting problem.

5.3. Universality of Arithmetic

Perhaps most profoundly, Gödel numbering shows that the simple operations of arithmetic are powerful enough to encode any formal system, suggesting a kind of universality in mathematics itself.

6. Conclusion

GoQuiche stands at an interesting intersection of mathematics, computer science, and formal logic. While it’s not a practical tool for everyday programming, it elegantly demonstrates the principle that programs are, at their core, mathematical objects that can be represented in purely numerical form.

The project serves as a reminder that there are deep connections between seemingly disparate fields. The same principles that Gödel used to explore the limits of formal systems can be applied to transform Go code into massive integers and back again. How fun!

In a field where practical utility often takes center stage, sometimes it’s valuable to explore mathematical curiosities like Gödel numbering—if only to appreciate the elegant theoretical foundations upon which computer science is built. And if nothing else, GoQuiche serves as a wonderful conversation starter about the relationship between code, mathematics, and the fundamental nature of computation.

Plus, there’s a certain satisfaction in creating a program whose output number is so large that if you tried to store it on conventional hardware, you’d need to explain to your cloud provider why you suddenly need an exabyte of storage for a single variable. I’d love to see that support ticket.

7. References

Bibliography

1. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38(1), 173-198.
   URL: https://www.jstor.org/stable/1968621 ↩
2. Nagel, E., & Newman, J. R. (2001). Gödel’s proof. NYU Press.
   URL: https://books.google.com/books?id=hXXyLQfAYl8C ↩
3. Hofstadter, D. R. (1999). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books.
   URL: https://books.google.com/books?id=o8jzWF7rD6oC ↩

Footnotes

1. Although we could improve efficiency by using a different encoding scheme, such as prime factorization without exponentiation, these alternative approaches would lose the elegant mathematical properties that make Gödel numbering theoretically valuable. But then again, if you’re trying to use this for practical purposes, you might need to reconsider your definition of “practical” anyway. ↩