Abstract:
Consider the language L with a single binary relation symbol <. A linear order,
A = ⟨A, <A⟩ is an L-structure so that <A orders A linearly i.e. <A is a transitive,
antireflexive relation so that for all x and y we have x <A y or y <A x or x = y.
Linear orders abound across mathematics - in particular
ℕ, ℤ, ℚ and ℝ are all first understood as linear orders. Amongst these, the rational order ℚ is
perhaps the most special. Every countable dense linear order without endpoints is isomorphic to ℚ (a theorem due to Cantor). Even more striking every countable linear order embeds into
ℚ. The goal of the lecture is first to explore these ideas a little further, proving Cantor’s theorem and the
generalization listed above and use them to conclude some interesting consequences in logic, particularly about definability and second, look at how these logical considerations can be applied to other areas of mathematics - particularly real analysis and topology.