[FOM] Characterization of the real numbers
JoeShipman@aol.com
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.
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
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