Understanding the Length of a Proof: The Case of 1 + 1 = 2

The equation \(1 + 1 = 2\) is one of the most fundamental truths in mathematics, often taken as an axiom rather than proven. However, if we were to consider constructing a formal proof for this statement within a logical framework, such as Peano arithmetic, we might be interested in how long such a proof could be.

Formal Proof and Axioms

In formal logic, a proof is a sequence of statements that are either axioms or logically follow from previous statements according to certain rules. For the statement \(1 + 1 = 2\), the Peano axioms provide a foundation. The Peano axioms define the natural numbers and their operations, including addition.

Peano Axioms

• Zero is a number: \(\exists x (x = 0)\)
• Successor function: Each number has a successor: \(\forall x (\exists y (y = S(x)))\)
• No two different numbers have the same successor: \(\forall x \forall y (S(x) = S(y) \rightarrow x = y)\)
• Zero is not the successor of any number: \(\forall x (x \neq 0 \rightarrow \exists y (y = S(x)))\)
• Addition: Addition is a function on pairs of numbers: \(\forall a \forall b (\exists c (c = a + b))\)

Proving \(1 + 1 = 2\)

To prove \(1 + 1 = 2\) using the Peano axioms, we need to define the numbers and the operation of addition within this framework. This process is non-trivial and typically involves several steps:

1. Define Numbers: Define (0, S(0), S(S(0)), \ldots) as natural numbers.
2. Define Addition: Define addition recursively:
3. (a + 0 = a)

4. (a + S(b) = S(a + b))

5. Prove Induction: Prove that if a property holds for zero and is preserved by the successor function, it holds for all natural numbers.

6. Compute (1 + 1):

7. By definition, (1 = S(0)).

8. Therefore, (1 + 1 = S(1) = S(S(0))).

9. Conclusion: Since (2) is defined as (S(S(0))), we have shown that (1 + 1 = 2).

Length of the Proof

The length of such a proof depends on how formal and detailed one wants to be. In a very formal proof, each step would need to be justified by an application of a specific axiom or inference rule. A proof assistant might generate a sequence of steps that could be hundreds of lines long, especially if every detail is spelled out.

However, in practice, mathematicians often use informal proofs and rely on established results from formal systems like Peano arithmetic. The key insight is that \(1 + 1 = 2\) follows logically from the axioms, regardless of how many steps are needed to reach this conclusion.

Conclusion

The equation \(1 + 1 = 2\) is a fundamental truth that can be proven within formal systems like Peano arithmetic. The length of such a proof can vary depending on the level of formality required, but it is ultimately justified by the foundational axioms of arithmetic.

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