[FOM] Characterization of the real numbers

JoeShipman@aol.com JoeShipman at aol.com
Sun Feb 6 15:23:22 EST 2005

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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