It's a technique that assigns unique natural numbers (Gödel numbers) to symbols, formulas, and proofs within a formal language, like arithmetic or logic.
This makes it possible to represent mathematical statements and arguments as numbers, enabling their manipulation within arithmetic itself.
Key steps involved:
1) Assigning numbers to symbols:
Each symbol in the formal language is assigned a unique prime number.
For example, 2 might represent "0", 3 might represent "=", 5 might represent "(", etc. Gödel numbering symbols
2) Gödel numbering symbols
Encoding sequences of symbols as numbers:
To represent a sequence of symbols (like a formula), Gödel numbers are combined using exponentiation.
Each symbol's prime number is raised to a power corresponding to its position in the sequence.
The resulting powers are then multiplied together to create a unique Gödel number for the entire sequence.
3) Gödel numbering formula example
Example:
The formula "0 = 0" would be encoded as:
Gödel number for "0": 2
Gödel number for "=": 3
Gödel number for "0": 2
Combining using exponentiation and multiplication: 2^1 * 3^2 * 2^3 = 2430
4) Significance:
Encoding of mathematics within mathematics: Gödel numbering allows statements about mathematics to be expressed as mathematical statements themselves.
Proving incompleteness theorems: Gödel used this technique to prove his famous incompleteness theorems, which demonstrate fundamental limitations of formal systems.
5) Other applications:
Gödel numbering has applications in computability theory, recursion theory, and other areas of logic and mathematics.
Key takeaways:
Gödel numbering provides a way to bridge mathematics and numbers.
It was crucial for proving fundamental limits of formal systems.
Its impact extends beyond logic into computer science and mathematics.
