Which is the axiom: well ordering principle, principle of induction, both, or none?

Question

From analysis 1 by Terence Tao, I learn that the principle of induction is a peano axiom. In many other analysis books, like analysis by Bartle and Sherbert, the well ordering principle is used to derive the principle of induction. I am thinking now: either both are equivalent statements or one of them is the axiom.

Answer 1

Strictly speaking it's an axiom schema, i.e. an infinite family of axioms, one for each unary predicate. For example, there's one for the predicate $\sum_{k=1}^n=\tfrac12n(n+1)$. There's one for $\sum_{k=1}^n=\tfrac12n(n+1)+1$, but only the former has a true base step, so, unlike the latter, it allows a proof by induction.

Anyway, the basic answer to your question is that which is the axiom (schema) is a matter of convention. Just about every text will take induction to the be the schema presented in a definition of the Peano axioms.

Something similar happens in set theory, where the axiom of choice is equivalent in ZF to all sets having a well-ordering. Combining either with ZF gives ZFC. The C is short for choice, so you can guess which version of the axiom (it's not a schema this time) is added - again, conventionally.

Answer 2

The answer, unfortunately, is that it depends on the circumstances. In some cases, the two statements are equivalent (and therefore either one could properly function as the fifth Peano axiom). In other cases, induction is stronger than the well-ordering principle in the sense that induction implies well-ordering, but not vice versa.

This is expounded upon more fully in the wonderful article:

Öhman, L. Are Induction and Well-Ordering Equivalent?. Math Intelligencer 41, 33–40 (2019). https://doi.org/10.1007/s00283-019-09898-4

which can briefly be summarized by his two conclusions:

(A) The induction principle and the well-ordering principle are not equivalent relative to the common first 4 axioms of the Peano system, since the resulting axiomatic systems admit different models.

(B) In the axiomatic system consisting of the common first 4 axioms together with the well-ordering principle, induction cannot be a theorem, since there is a model Ord ($\omega+\omega$) in which all five of these axioms are satisfied, but induction is not true.

He goes on to state:

We may also note that the axiomatic system consisting of the Peano axioms with induction admits only models that are isomorphic to the natural numbers, and since the natural numbers are well-ordered, in this system well-ordering principle is in fact a theorem. Induction is therefore stronger than well-ordering in this context, in that it has the power to rule out more possible models.