I know that one characterization of the reals is that it is the only Dedekind-complete ordered field. Are there any other characterizations of the reals as a field?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
3
Here are a few I know off hand:
The reals are the unique dense linear order without endpoints that is both Dedekind complete and separable.
The reals are the terminal Archimedean ordered field.
The real compact interval can be constructed uniquely as a terminal co-algebra for a certain “wedge” functor on the category of bi-pointed topological spaces.
-
1$\begingroup$ @mathlander: Answers to the OP's question depend on what is meant by "the reals". For example, see this question and this other question for topological characterizations of the reals. $\endgroup$ Commented Oct 27, 2022 at 19:02
-
1$\begingroup$ For another example (thought of while shaving just now), as a set the reals are characterized by having cardinality $\mathfrak c.$ $\endgroup$ Commented Oct 27, 2022 at 19:24
-
$\begingroup$ I want the real numbers, not the real line. $\endgroup$ Commented Oct 27, 2022 at 20:44