Question

Construct a function ℕ --> ℕ other than the usual successor function which satisfies Peano's Axioms.

Answer 1

The question aims to construct a function from the set of natural numbers to itself other than the successor function, satisfying Peano's axioms. The provided example is the doubling function f(n) = 2n, which satisfies these axioms through verification of each axiom individually.

Step-by-step explanation:

In mathematics, specifically in the branch of number theory, Peano's axioms are a set of axioms for the natural numbers (ℕ). A well-known function that satisfies these axioms is the successor function, S(n) = n + 1. However, to construct another function from ℕ to ℕ that satisfies Peano's axioms, consider the function f(n) = 2n. This doubling function takes a natural number n and maps it to its double 2n, which is also a natural number.

Let's verify that f(n) satisfies Peano's axioms:

1. 0 is a natural number.
2. For every natural number n, f(n) is a natural number. (Since the double of a natural number is also a natural number)
3. For all natural numbers m and n, if m ≠ n, then f(m) ≠ f(n). (Doubling is an injective function)
4. The number 0 is not in the image of f. (Since there is no natural number n such that 2n = 0 in ℕ)
5. The last axiom regarding induction can also be satisfied, provided that the mathematical induction is defined properly in the context of f(n).

The function f(n) = 2n is thus an example of a function that satisfies Peano's axioms other than the successor function.