Induction over the Continuum
We exploit the analogy between the well ordering principle for nonempty subsets of N (the set of natural numbers) and the existence of a greatest lower bound for non-empty subsets of [a,b)1 to formulate a principle of induction over the continuum for [a,b) analogous to induction over N. While the gist of the idea for this principle has been alluded to, our formulation seems novel. To demonstrate the efficiency of the approach, we use the new induction form to give a proof of the compactness of [a,b]. (Compactness, which plays a key role in topology, will be briefly discussed.) Although the proof is not fundamentally different from many familiar ones, it is direct and transparent. We also give other applications of the new principle.
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