# [FOM] Characterization of the real numbers

John Baldwin jbaldwin at uic.edu
Sun Feb 6 21:52:25 EST 2005

```I think a  standard characterization  is a complete linear order with
countable dense subset.
Whether countable dense subset could be replaced by any set of disjoint
intervals is countable yields the Suslin problem.

This is a full 2nd order axiom. For attempts to get it in more
restricted
logics see

pages 106 and especially 19.11 on page 109 of my monograph
at

http://www2.math.uic.edu/~jbaldwin/model.html

(first article in list.)  Sorry I messed up the reference to the
Baumgartner article. I will try to get that fixed but I don't have it at
hand.

On
Sun, 6 Feb 2005 JoeShipman at aol.com wrote:

> What is the simplest characterization of the real numbers?  That is, what is the simplest description of a structure, any model of which is isomorphic to the real numbers?
>
> A standard way of characterizing the real numbers is "ordered field with the least upper bound property".  But do I need to refer to field operations?  "Dense ordering without endpoints and the least upper bound property" isn't sharp enough.  "Homogenous dense ordering with the least upper bound property" looks better, except that it doesn't rule out the "long line" (product of the set of countable ordinals with [0,1} in dictionary order, with intial point removed).  (It also doesn't rule out the reversed long line, or the symmetric long line.)
>
> The best I can do without referring to relations other than the order relation is "dense ordering with least upper bound property, isomorphic to any of its nonempty open intervals". Can anyone improve on this?
>
> -- JS
>
>
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John T. Baldwin
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Department of Mathematics, Statistics, and Computer Science
jbaldwin at uic.edu
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