prove 1+1 = 2 without using the fact 1+1 = 2

Proving that \(1 + 1 = 2\) without directly assuming it requires a foundational approach, usually grounded in formal logic and set theory. One such formal proof comes from **Peano's axioms**, which define the natural numbers and their arithmetic. Here’s an outline of the proof:

### Step 1: Define the Natural Numbers
Using Peano's axioms, we define natural numbers:

1. **0 is a natural number.**
2. **Every natural number \( n \) has a successor, denoted as \( S(n) \), which is also a natural number.**

We can represent the first few natural numbers as:
- \( 0 \)
- \( S(0) \) (which we call 1)
- \( S(S(0)) \) (which we call 2)
- And so on.

### Step 2: Define Addition
Addition is defined recursively:

1. For any natural number \( n \), \( n + 0 = n \).
2. For any natural numbers \( m \) and \( n \), \( m + S(n) = S(m + n) \).

### Step 3: Define 1
By Peano's axioms, \( 1 \) is the successor of 0, i.e., \( 1 = S(0) \).

### Step 4: Prove \( 1 + 1 = 2 \)
Using the definition of addition:

1. Start with \( 1 + 1 \).
2. Since \( 1 = S(0) \), replace it to get \( S(0) + 1 \).
3. By the definition of addition, \( S(0) + 1 = S(S(0) + 0) \).
4. By the first rule of addition, \( S(0) + 0 = S(0) \).
5. Thus, \( S(S(0) + 0) = S(S(0)) \).

Finally, by the definition of 2, \( S(S(0)) = 2 \).

Therefore, \( 1 + 1 = 2 \).

### Summary
The proof relies on defining the natural numbers and addition formally through Peano's axioms. The result \( 1 + 1 = 2 \) is derived through these definitions, without assuming the outcome in advance.

Would you like more details or have any questions?

Here are 5 related questions:

1. What are Peano's axioms in detail?
2. How do we define other basic arithmetic operations using Peano's axioms?
3. Can we derive multiplication from the same set of axioms?
4. How does set theory relate to the proof of basic arithmetic operations?
5. What are the limitations of Peano's axioms in arithmetic?

**Tip:** Understanding how basic arithmetic is constructed from foundational axioms helps in grasping more complex mathematical theories.

Explore the proof of 1 + 1 = 2 without assuming the result using Peano's axioms and formal logic. Understand the foundational concepts of set theory and formal logic applied in mathematics.

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