Weakly compact cardinals and nonspecial aronszajn trees

LEMMA 1. If λ i;…ardinal with cf λ > ω, then □x implies that there i;…#x03BB;₊-Aronszajn tree with an ω-ascent path, i.e;…equence (x^α: α < λ⁺) with each x_α;…x^α_n;…#x003C; ω…ne-to-one sequence from T_α, such that for all α < β < λ⁺, x^α_n precedes x^β_n, in the tree order for sufficiently large n. LEMMA 2. If λ i;…ardinal with cf λ…#x03C9; < λ, then □x implies that there i;…#x03BB;⁺-Aronszajn tree with an ω₁-ascent path (replace ω by ω_i, above). LEMMA 3. If λ is an uncountable cardinal;…s regular;…#x003C; λ, cf λ ≠ k;…s aλ⁺ -Aronszajn tree, and (x^α_i;…#x003C; k) i;…ne-to-one sequence from T_ζ(α) with the property of ascent paths, where ζ: λ⁺ ⟶λ⁺ i;…onotone increasing function of α, the;…s nonspecial. THEOREM 4. If λ is uncountable, then □_λ implies that there i;…onspecial λ⁺-Aronszajn tree. THEOREM 5. If λ is an uncountable cardinal;… λ⁺, an;…s not (weakly compact)^L, then there i;…onspecial K-Aronszajn tree.