As examples here we define the natural numbers, of course here I use the notational set theory, so I use really from Neuman naturals. So instead of writing plus I add this because I am here modelling everything in set theory. Here is the definition of Fibonacci as the graph of Fibonacci is the least fixed point of a given operator. And we illustrate the power of induction by showing that Fibonacci as we defined it earlier in the lecture is equal to Fibonacci as defined here. And you see that all the idea is to invent a new set which satisfies. It is a prefixed point for the operator to define natural numbers and since we know that natural numbers are the least such set well then we know that the natural numbers are included in this predicate, in this subset. And of course, this implies that the two are equal.

---