= No categorical cardinal for the complete named rational diagram
For any infinite cardinal $\kappa$, insert a dense order without endpoints of size $\kappa$ into an irrational cut of the named rationals. Compare it with an order of the same size having a dense tail above every named rational. Both <first-order model>[models] satisfy the complete named rational diagram by <quantifier elimination>, but only the second realizes the type of being above every named rational. This is an <isomorphism of structures>[isomorphism] invariant over the constants, even though it is an infinite type rather than one defining sentence. Thus the diagram is categorical in no cardinal with a <first-order model>[model].
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