Is there a useful application of Peano arithmetic? If there is, can someone provide an example of how Peano arithmetic can be used to solve a real-world problem?
If not, can someone provide an example of any axiomatic system other than ZFC that can be used to solve a real-world problem?
 A: The setting out of the axioms of Peano Arithmetic in the 19th century was just one attempt to determine the essential properties of the natural numbers from which all other properties might be derived. In a sense, PA and other such attempts were reverse-engineered from "real-world" problems. To my knowledge, they have not resulted in any new solutions to real-world problems, but that wasn't their purpose. 
A: When I balance my check book, I am using integer arithmetic (because I can do everything in cents).  When I get the balance from the bank at the end of each month, and compare it to my running balance I keep in my check book, I have to account for uncleared checks and things like that.  In order to do this, I end up adding and subtracting the numbers in a different order than the bank did.  That is, I use the fact that integer arithmetic is commutative and associative.  And how do I know this?  Because I can prove this using Peano's axioms.
I have to say that my faith in the Peano axioms is exceedingly great.  Because whenever the numbers in my bank balance don't work, I automatically presume that either I or the bank have made a mistake.  I never consider the possibility that I might have discovered an inconsistency in arithmetic.  And this is despite Goedel's Theorem that says I can never be sure.
