[FOM] Characterization of the real numbers

[JoeShipman@aol.com]

Yes, but "having a countable dense subset" involves significantly higher logical complexity, you have to define "countable subset".  I'd like to avoid defining the integers if possible.

Mayberry wrote:
How about "complete linear ordering without endpoints having a countable 
dense subset". I think this is Cantor's characterization.

--On 06 February 2005 15:23 -0500 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