Question

From the Wikipedia entry on recursive definition:

An inductive definition of a set describes the elements in a set in terms of other elements in the set. For example, one definition of the set N of natural numbers is:

1. 1 is in N.
2. If an element n is in N then n+1 is in N.
3. N is the smallest set satisfying (1) and (2).

There are many sets that satisfy (1) and (2) – for example, the set {1, 1.649, 2, 2.649, 3, 3.649, …} satisfies the definition.

I don’t understand why (3) is needed. In the example given, it states that 1.649 is a member of this set but 1.649 doesn’t satisfy (1) or (2).

Why is (3) needed and how is 1.649 in the set?

Answer 1

Rule 2 on the list is an “if”, not an “if and only if”. They’re not rules for generating the set, they’re rules for deciding if a set is allowed. The set {1, 1.649, 2, 2.649, 3, 3.649, …} satisfies rule 1, because 1 is in the set. It satisfies rule 2, because for every element of the set, that element plus one is also in the set. In fact, even the set of real numbers satisfies the first two rules, and it has uncountably many “extra” elements that you don’t need.

Only rule 3 stops you adding arbitrary extra elements to the set, by saying the set has to be the smallest possible one.