## Infectious zero spreads to all natural numbers

This is like saying a set of naturals closed under succession that contains zero equals all natural numbers. A set of naturals we could call a natural set.

Infected zero equals all naturals. If by infected we mean also infectious itself.

Infections on finite sets can also be used to teach mathematical induction. If one student gets the flu, the whole class does. Although, it can branch in such an infection.

For natural numbers, we only allow infection in one direction and only to the successor. Or at least, that is the setup for standard mathematical induction.

Course of values induction is where the entire initial segment up to its end infects the next natural. So an initial segment is 0 to n. We can call it the head segment n. The successor to a head segment we can use to mean n’. So the head segment n infects n’ is the course of values induction step. We can call this the infection step.

Infection proofs.

1. Show 0 is infected.  Infection base.
2. Assume n is infected.  Infection hypothesis.
3. Prove that n’ is infected.  Infection step.
4. Conclude that all naturals are infected.  Conclusion step.

Infected means a proposition is true, or a function is defined.