I am getting introduced to mathematical induction, and I am trying to formulate induction principles for various domains. I have been asked to "give an induction principle for the integers." As an example, the standard (weak) induction principle for the natural numbers is:
"If 0 is P and, if a natural number n is P, then its successor (i.e. n+1) is also P, then all natural numbers are P."
I'm uncertain about this one because there's not an obvious base case for the integers (since minus infinity is not a number, we can't let that be the base case). My intuition is telling me that the solution will require a strong principle of induction to get around there not being an identifiable smallest element. Can anyone help?