To prove that \(1 + 1 = 2\), we can rely on the foundational principles of mathematics, specifically Peano's axioms. These axioms define the natural numbers and basic arithmetic operations. Here's a step-by-step outline:

### Peano's Axioms (Simplified)
1. **0 is a natural number.**
2. **Every natural number \(n\) has a successor, denoted by \(S(n)\), which is also a natural number.**
3. **0 is not the successor of any natural number.**
4. **Different numbers have different successors, i.e., if \(S(n) = S(m)\), then \(n = m\).**
5. **A property that holds for 0 and holds for the successor of any number that it holds for, holds for all natural numbers (induction).**

### Proof that \(1 + 1 = 2\)

1. **Define 1 and 2:**
   - The number \(1\) is defined as the successor of \(0\), i.e., \(1 = S(0)\).
   - The number \(2\) is defined as the successor of \(1\), i.e., \(2 = S(1)\).

2. **Define Addition:**
   - Addition can be defined recursively using Peano's axioms.
   - For any natural number \(a\): \(a + 0 = a\).
   - For the successor of any number \(b\): \(a + S(b) = S(a + b)\).

3. **Apply the Definition of Addition:**
   - Consider \(1 + 1\):
   - By definition, \(1 = S(0)\).
   - Therefore, \(1 + 1 = 1 + S(0)\).

4. **Use the Recursive Definition:**
   - Using the recursive definition: \(1 + S(0) = S(1 + 0)\).
   - Since \(1 + 0 = 1\) by the base case of addition, \(S(1 + 0) = S(1)\).

5. **Identify the Result:**
   - \(S(1)\) is by definition \(2\).
   - Hence, \(1 + 1 = S(1) = 2\).

Thus, \(1 + 1 = 2\) is proven using Peano's axioms.

Would you like more details or have any questions? Here are five related questions:

1. What are Peano's axioms in full detail?
2. How is addition defined for natural numbers?
3. What is the significance of using a formal proof?
4. Can \(1 + 1 = 2\) be proven in other number systems?
5. How do axioms serve as the foundation of arithmetic?

**Tip:** Understanding the formal definitions and axioms behind basic arithmetic operations provides a deeper insight into the structure of mathematics.

Explore the proof that 1 + 1 = 2 using Peano's Axioms with a detailed step-by-step explanation. Understand the foundational principles of arithmetic and recursive definitions involved in proving this statement. Suitable for advanced high school students and anyone interested in mathematical proofs.

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