Definition 3.3 (Theory of dense linear orders). Let $L=\{<\}$. We define the theory in axioms:

• (i)
  Irreflexive: $\forall x,\neg (x<x)$.
• (ii)
  Transitive: $\forall x,\forall y,\forall z,((x<y\wedge y<z)\to x<z)$.
• (iii)
  Antisymmetric: $\forall x,\forall y,(x\ne y\to (x<y\vee y<x))$.
• (iv)
  Dense: $\forall x,\forall y,(x<y\to (\exists z(x<z<y))$.
• (v)
  No endpoints: $\forall x,\exists y,\exists z,(z<x<y)$.