# 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.

Axioms:

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).

## Solution

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