According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that $0$ is in the set and for every real number $x$ in the set, $x + 1$ is also in the set and a natural number is a real number that every inductive set contains. The problem with that definition is that it is circular because the real numbers are constructed from the natural numbers.
Answer
Suppose we did start with some notion of "natural number" which we used to construct a model of the real numbers.
Then even in this setting, the quoted definition is still not circular, because it's defining a new notion of "natural number" that will henceforth be used instead of the previous notion of "natural number".
We could give the new notion a different name, but there isn't really any point; the new version of "natural numbers" has an obvious isomorphism with the old version so it's not really any different from the old one in any essential way.
There are a number of reasons why an exposition of real analysis might construct the natural numbers from the real numbers; the two most prominent are:
- It is technically convenient to have the natural numbers be a subset of the real numbers
- It makes the exposition somewhat more agnostic about foundations; it simply needs the real numbers as a starting point
No comments:
Post a Comment