First-Order Continuous Induction, and a Logical Study of Real Closed Fields

Over the last century, the principle of "induction on the continuum" has been studied by different authors in different formats. All of these different readings are equivalent to one of the three versions that we isolate in this paper. We also formalize those three forms (of "continuous induction") in first-order logic and prove that two of them are equivalent and sufficiently strong to completely axiomatize the first-order theory of the real closed (ordered) fields. We show that the third weaker form of continuous induction is equivalent with the Archimedean property. We study some equivalent axiomatizations for the theory of real closed fields and propose a first-order scheme of the fundamental theorem of algebra as an alternative axiomatization for this theory (over the theory of ordered fields).