Set definitions have to be precise

A set of rules cannot diverge to 2 different sets.

Scenario 1

An example is N, set of all natural numbers.


Zero is in the Set

Succ(n) is in the set, if n is in the set.

Then, Num = {Zero, Suc(Zero), Suc(Suc(Zero)), …} satisfies this.

However, StrangeNum also satisfies this = {Zero, Suc(Zero), …} ∪ {X, Suc(X)}, it contains 0, all elements are Suc(n).


As such, we need to have “least” set, smallest with respect to the subset ordering on sets. i.e. no extra stuff.

This allows us to use inductive principle:

Suppose l is inductively defined by rules R.

Show that every x ∈ l has property P, it is enough to show that P satisfies the rules of R.

In the above scenario, we can drop {X, Suc(X), …}.

Scenario 2

Define a tree. Does every tree have a height

We can have infinite Trees -> No

We need to use “least set” to prevent infinite Trees -> Finite Trees -> Every tree has height