Version of the Axiom of Induction for Real Induction?

Question

Mathematical induction can be done using the axiom of induction, which is given as a formula written in the language of mathematical logic. Is there a way to express the ideas behind 'real induction' in mathematical logic as well?

Answer 1

I believe that Theorem 3 of that paper, when unwound, is what you are looking for. That theorem just says that whenever a totally ordered set is Dedekind complete, then its only inductive subset is the whole thing. (Actually, it says the converse too, but I believe this direction is the one that interests you.)

Do you really want to see real induction written down as a sentence in the language of set theory? (The language of arithmetic isn't suitable for RI since RI talks about arbitrary total orderings.) This can be done, but it's uuuglyyy . . .

To give a sense of what I mean by "ugly," here's the predicate "$X$ has the greatest upper bound property" written out entirely in symbols:

$$\forall A(\exists y\forall a(a\in A\implies a<y)\implies\exists y(\forall a(a\in A\implies a<y)\wedge \forall z(z<y\implies \exists b\in A(z<b))).$$

(Quantifiers here range over $X$ or $\mathcal{P}(X)$ as is appropriate.) And this is just one tiny piece of the principle of real induction: we would need to translate the whole statement "for total orders, Dedekind complete iff principle of ordered induction" into symbols.

The point is: since induction on arbitrary totally ordered sets is much more general than induction on $\mathbb{N}$ alone, we should expect the statement of the former to be much longer than the statement of the latter; in particular, we shouldn't be surprised if it can't be written in one short string of symbols.

Answer 2

Your question can be interpreted in various ways. In a sense, the axiom of induction for cardinalities larger than countable is simply the axiom of choice, in the form of transfinite induction (or recursion as it is sometimes referred to). Along these lines you may find Transfinite induction on the real line to be of interest. It discusses numerous modern application of induction on the real numbers.