Well-orders are not first-order axiomatizable
= Well-orders are not first-order axiomatizable
If a theory axiomatized the <well-orders>, adjoin constants $c_0,c_1,\ldots$ and the sentences $c_{n+1}<c_n$. Every finite fragment has a finite well-ordered model, but <compactness theorem>[compactness] would produce a model with an infinite descending chain.