Godel Numbering in Automata Theory

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In meta-mathematics and logics, Gdel Numbering is an interesting concept. It was developed by Austrian mathematician Kurt Gdel. This concept plays a huge role in Gdel's incompleteness theorems. In this chapter, we will see the basics of Gdel Numbering, explain how it works, and walk through a couple of examples for a better understanding.

Gdel Numbering

Gdel Numbering provides a method of encoding mathematical expressions as numbers. This technique allows us to transform abstract mathematical concepts into arithmetic notations, which makes it possible to apply number theory to questions of logic.

Gdel developed this method as part of his proof that there are true statements in mathematics that cannot be proven within a given formal system. This idea was a groundbreaking result that challenged the previous efforts to create a complete and consistent set of mathematical axioms.

Understanding the Basics

Let us first understand a few basic concepts in logic and mathematics −

• Calculus − In this context, calculus refers to a system of signs that do not have any inherent meaning on their own. These signs are used to construct formulas.
• Variables − Variables can be either sentential (like p, q, r) or numerical (like x, y, z). Sentential variables can be replaced by statements, while numerical variables can be replaced by numbers.
• Connectives − These are logical operators like "not" (~), "and" (), "or" (∨), and "if...then" (⊃).
• Quantifiers − Quantifiers such as "for all" (∀) and "there exists" (∃) are used to generalize statements.

Let us understand how Gdel Numbering is constructed.

Constructing Gdel Numbers

Gdels approach starts with the idea of assigning a unique number to every symbol, the formula, and the proof within a logical system. This unique number is known as a Gdel Number.

To achieve this, Gdel chose a system of symbols that includes all cardinal numbers and arithmetic operations like addition and multiplication. The idea is that each symbol and formula in the system can be translated into a unique number.

Basic Symbols of Gdel Numbers

Here is how Gdel assigned numbers to some basic symbols −

For example, the symbol "~" (which means "not") is assigned the Gdel Number 1, and the symbol "V" (which means "or") is assigned the Gdel Number 2.

Defining Variables

Next, we have to define variables using Gdel Numbers. There are two types of variables −

• Numerical variables (x, y, z) and
• Sentential variables (p, q, r).
• Predicate variables are also used, representing statements like "is prime."

Let us see how these variables might be assigned Gdel Numbers −

Example of Gdel Numbering

Let us see an example to see how Gdel Numbering is applied. Consider the expression (∃x) (x = sy), which means "there exists an x such that x is the immediate successor of y."

The Gdel Numbers for each symbol in this expression are −

• ( : 8
• ∃ : 4
• x : 13
• ) : 9
• = : 5
• s : 7
• y : 17

Now, instead of representing this formula as a sequence of numbers, Gdel proposed converting it into a single unique number. This is done by raising the first n prime numbers to the power of the corresponding Gdel number and then multiplying the results together.

The calculation for our example would look like this −

$$\mathrm{2^8 \times 3^4 \times 5^13 \times 7^9 \times 11^5 \times 13^7 \times 17^17}$$

This huge number, which we can m, is guaranteed to be unique due to the Fundamental Theorem of Arithmetic (which states that every number has a unique prime factorization).

Extending Gdel Numbering to Proofs

Gdel's method also allows us to assign unique numbers to entire proofs, which are sequences of formulas. If we have two formulas with Gdel Numbers m and n, the Gdel Number for the proof that links them would be calculated as −

$$\mathrm{k = 2^m \times 3^n}$$

This method shows that every symbol, formula, and proof in the system can be uniquely identified by a Gdel Number.

Decrypting Gdel Numbers

We have seen how we can convert the expression to Gdel Numbers, let us see the decoding now. We can work backward to determine what expression it represents. For example, if we are given the number 243,000,000, which has the prime factorization 2⁶ × 3⁵ × 5⁶, we can decode it to find the original expression it represents. In this case, it corresponds to the expression 0=0.

Arithmetization of Meta-Mathematics

This idea was unique and has several applications. Gdel's next step was to express statements about mathematical formulas as arithmetic relationships between their corresponding Gdel Numbers.

For example, if a formulas Gdel Number is a, and we want to know what the first symbol in the expression is, we can examine the prime factorization of a. The exponent of the first prime in the factorization reveals which symbol it is.

Conclusion

In this chapter, we have covered the basics of Gdel Numbering. This is a system that encodes mathematical expressions as unique numbers. We explored the fundamentals of this concept, how Gdel Numbers are assigned, with an example to how it works.

We also explained how Gdel's method can be extended to proofs and used to decipher expressions from their Gdel Numbers.