To prove that \(1 + 1 = 2\), we can use the foundational principles of arithmetic and number theory. One of the most well-known formal systems that provides such a proof is *Peano Arithmetic*. Below is an outline of the proof using the Peano axioms:

### Peano Axioms (Simplified)
1. **Zero is a number.**
2. **Every natural number has a successor.** (If \(n\) is a natural number, then there exists a natural number called the successor of \(n\), denoted by \(S(n)\).)
3. **Zero is not the successor of any natural number.**
4. **Different numbers have different successors.** (If \(m\) and \(n\) are different, then \(S(m)\) and \(S(n)\) are different.)
5. **If a set contains zero and the successor of every number in the set, then the set contains all natural numbers.** (This is the principle of mathematical induction.)

### Defining Numbers:
- \(1\) is defined as the successor of \(0\), i.e., \(1 = S(0)\).
- \(2\) is defined as the successor of \(1\), i.e., \(2 = S(1)\).

### Addition Operation:
- Addition is defined recursively as follows:
  - For any natural number \(n\), \(n + 0 = n\).
  - For any natural numbers \(m\) and \(n\), \(m + S(n) = S(m + n)\).

### Proof that \(1 + 1 = 2\):
1. **Start with the definition of 1:** \(1 = S(0)\).
2. **Apply the addition definition:**
   \[
   1 + 1 = S(0) + 1
   \]
3. **Apply the recursive definition of addition:**
   \[
   S(0) + 1 = S(0) + S(0)
   \]
   Using the addition rule \(m + S(n) = S(m + n)\), we substitute \(m = S(0)\) and \(n = 0\):
   \[
   S(0) + S(0) = S(S(0) + 0)
   \]
4. **Simplify using \(n + 0 = n\):**
   \[
   S(S(0) + 0) = S(S(0))
   \]
   Since \(S(0) = 1\), it follows:
   \[
   S(S(0)) = S(1) = 2
   \]
5. **Conclusion:**
   \[
   1 + 1 = 2
   \]

This completes the formal proof that \(1 + 1 = 2\) using the Peano axioms.

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Would you like more details or have any questions?

### Related Questions:
1. How are the Peano axioms foundational in mathematics?
2. What is the significance of the successor function in number theory?
3. How does mathematical induction work in proving statements?
4. Can you prove that \(2 + 2 = 4\) using a similar approach?
5. What are other formal systems that can prove \(1 + 1 = 2\)?

### Tip:
Understanding the Peano axioms is crucial for grasping the fundamentals of number theory and arithmetic operations.

Discover a formal proof that 1 + 1 equals 2 using Peano axioms and mathematical induction. This detailed explanation covers foundational principles in number theory, including the recursive definition of addition. Suitable for students in Grades 10-12.

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