Lecture Notes on Mathematical Induction
نویسنده
چکیده
This is not a proof. (No good proof uses “and so forth” to gloss over a key point!) But the idea is as follows: we can keep iterating the above argument as many times as we want, deducing at each stage that since S contains the natural number which is one greater than the last natural number we showed that it contained. Now it is a fundamental part of the structure of the positive integers that every positive integer can be reached in this way, i.e., starting from 1 and adding 1 sufficiently many times. In other words, any rigorous definition of the natural numbers (for instance in terms of sets, as alluded to earlier in the course) needs to incorporate, either implicitly or (more often) explicitly, the principle of mathematical induction. Alternately, the principle of mathematical induction is a key ingredient in any axiomatic characterization of the natural numbers.
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