= Cardinality of the family of strict total orders
{title2=$|\mathrm{LO}(\kappa)|=2^\kappa$}
For an infinite well-orderable <cardinal number> $\kappa$, each strict total order is a subset of $\kappa^2$, giving at most $2^\kappa$ possibilities by the <square of an infinite cardinal>. For a lower bound, order $\kappa$ two-element blocks by the ordinal index and independently reverse each block according to a subset of $\kappa$. Transport along $\kappa\times2\cong\kappa$. Distinct subsets give distinct orders. The usual <axiom of choice> development supplies these product identities; for well-orderable cardinals they are already provable in <ZF>.
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