Is Real Numbers an Ordered Field Under <?

[ashok vardhan]

Sir,i have read in wikipedia that for a relation to be ordered it should be transitive,antisymmetric,total...however we know that Real numbers is an ordered field under relation "<" but antisymmetric property is not valid with "<" relation..how is this justified..rectify me...

 

[micromass]

The real numbers are not an ordered field under <.
Rather, the real numbers are an ordered field under $\leq$.