# [FOM] Characterization of the real numbers

Stephen Lavelle icecube at maths.tcd.ie
Mon Feb 7 16:06:48 EST 2005

```There is the categorical classification of the real interval as a termal
object in a certain category.  It's an elementary result - the simplest
explanation I've come across is Q17 in this problem set by Leinster, where
the details are explained

http://www.maths.gla.ac.uk/~tl/categories/sheetthree.ps

Hope that's of some help,

Stephen

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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```