[FOM] Characterization of the real numbers
JoeShipman at aol.com
Sun Feb 6 20:10:54 EST 2005
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.
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
More information about the FOM