Linear Orderings

a₁ a₂ additively indecomposable algorithm assume automorphism b₁ b₂ binary relation C₁ c₁(a cardinal number choose condensation construction contains continuum hypothesis Corollary countable linear ordering countable ordinal D₁ define DEFINITION denoted dense linear ordering dense order Ehrenfeucht embedding Erdös example Exercise finitary finite number finite subset formula G-equivalent given Hajnal Hence I₁ induction hypothesis infinite initial segment interval isomorphism k-partitioned ordering last element Lemma limit ordinal M₁ Math n-tuple n-type n₁ natural numbers order type order-preserving order-preserving map partition PLAYER predicate quantifier depth R₁ Ramsey's Theorem rational numbers recursive function recursive ordinal recursive set recursive subset recursively enumerable result satisfied saturated model scattered linear orderings Show Sierpinski statement subordering subset of Q successor successor ordinal Suppose T₁ uncountable v₁ w₁ well-ordering winning strategy τ₁