M ar 2 01 6 Structural Properties of R 2 Part II Timothy

This is the second of two papers establishing structural properties of R2, the structure giving rise to pure patterns of resemblance of order two, which partially underly the results in [2] and [3] as well as other work in the area. For the entire paper, we will assume ρ is an arbitrary additively indecomposable ordinal and α is an epsilon number greater than ρ. By Lemma 5.8 of [4], κα is additively indecomposable. The Order Reduction Theorem says that for η < θ 2(α), J ρ α,η is essentially isomorphic to an initial segment of R2 . The important point is that this reduction reduces the lengths of chains in ≤2: if the longest chain of elements of J α,η with respect to ≤2 has length n + 1 where n ∈ ω then the longest chain among the elements of the corresponding initial segment of R2 with respect to the analogue of ≤2 in R α 2 has length at most n. The Second Recurrence Theorem for ≤2 says roughly that the initial segment of R2 corresponding to Iωη grows by adding ρ new intervals of the form I β each time η increases. In the case ρ = 1, which is essentially R2, this says one new interval is added each time η increases. Section 1 contains background. Section 2 introduces the notion of local incompressibility. Section 3 establishes the Order Reduction Theorem. Section 4 establishes the Second Recurrence Theorem for ≤2.