Why is better to work with first-order Peano's axioms than with second-order Peano's axioms?

Question

In the case of Peano's axioms the second-order version is categorical, but the first-order is not. Besides, with the second-order version the operations: addition, multiplication and exponentiation can be defined, but with the first-order version they cannot. However, in books they prefer and promote the use the first-order version. Why is better to work with first-order Peano's axioms than with second-order Peano's axioms?

Answer 1

Each set of axioms has its own purposes, and neither is better than the other in all circumstances.

It is true that, when they are interpreted in set theory, the second-order Peano axioms are categorical. They are useful for characterizing the natural numbers once we have a notion of "set" to work with.

The second-order Peano axioms that most people think of are not so useful for reasoning about natural numbers. To be clear, I am referring to the following list of three axioms:

• $S(x) \not = 0$
• $S(x) = S(y) \to x = y$
• $(\forall X)[( 0 \in X \land (\forall n)[n\in X \to S(n) \in X]) \to (\forall n)[n \in X]]$.

The issue in working "with" these axioms is that they don't tell you how to construct any set $X$ to apply the third axiom. So, if you want to proceed on an axiomatic basis, you have to add additional axioms to allow for the construction of sets, before you can actually use the third axiom in any nontrivial way.

For example, the categoricity proof that is mentioned in the questions is not proved "from" these axioms - it is proved "about" these axioms in the metatheory. Similarly, the definition of the addition function mentioned in the question is not proved from the three axioms, it is proved in the metatheory using set-theoretic methods.

When you add set existence axioms to the three above, in order to start writing formal proofs from the axioms, what you will be able to prove is strongly influenced by which set existence axioms you add. For example, we can make the theory known as "second order arithmetic" ($\mathsf{Z}_2$) by adding set existence axioms which quantify only over numbers and sets of numbers, or we can make a stronger theory by adding all the set existence axioms of ZFC. Each of these systems is able to prove many things that the three axioms above cannot prove in the usual deductive system for predicate logic.

On the other hand, the axioms of Peano arithmetic (which are the axioms of a discrete ordered semiring, abbreviated $\mathsf{PA}^-$, plus the axiom scheme of induction) can be used "as is" to prove many nontrivial theorems. They are particularly useful when we want to prove things about the natural numbers without getting involved in set theoretic issues and without referring to set existence axioms.