@Powerful_Tiger

9 months ago

Steps to find 1+1

1. Consider two natural numbers, where the first number, denoted as \( a \), is equal to 1.
2. Similarly, let the second number, denoted as \( b \), also be equal to 1.
3. The operation to be performed is addition, symbolized by the plus sign (+), which is a binary operation that combines these two numbers.
4. According to the Peano axioms, which form the foundation of the natural numbers, the successor of a natural number \( n \) is denoted as \( S(n) \), representing \( n + 1 \).
5. Given that \( a = 1 \), we recognize that \( a \) is the successor of 0, i.e., \( 1 = S(0) \).
6. Similarly, \( b = 1 \), so \( b \) is also the successor of 0, i.e., \( 1 = S(0) \).
7. The addition of two natural numbers is defined recursively as follows:
- For any natural number \( n \), \( n + 0 = n \).
- For any natural numbers \( n \) and \( m \), \( n + S(m) = S(n + m) \).
8. Applying these definitions, the sum of \( a \) and \( b \) can be calculated step-by-step:
- Begin with \( a + b \).
- Substitute \( a \) and \( b \) with their values: \( 1 + 1 \).
- Recognize that 1 is the successor of 0: \( S(0) + S(0) \).
- By the recursive definition, \( S(0) + S(0) \) is equivalent to \( S(0 + S(0)) \).
9. Further simplifying, \( 0 + S(0) \) is \( S(0) \), as per the definition of addition with 0.
10. Thus, \( S(0 + S(0)) = S(S(0)) \).
11. By the property of successors, \( S(S(0)) \) is the successor of the successor of 0, which is the number 2.
12. Therefore, \( 1 + 1 = 2 \).

In summary, after considering the natural numbers, the definition of the successor function, and the recursive definition of addition, we can conclude that the sum of 1 and 1 is indeed 2.

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