Nonvanishing derived limits without scales

The derived functors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mo>lim</mml:mo> <mml:mi>n</mml:mi></mml:msup> </mml:math> of the inverse limit are widely studied for their topological applications, among which are some repercussions on the additivity of strong homology. Set theory has proven useful in dealing with these functors, for instance in the case of the inverse system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> of abelian groups indexed over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mmultiscripts><mml:mrow></mml:mrow> <mml:mrow></mml:mrow> <mml:mi>ω</mml:mi></mml:mmultiscripts> <mml:mi>ω</mml:mi></mml:mrow> </mml:math> . So far, consistency results for nonvanishing derived limits of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> have always assumed the existence of a scale (i.e. a linear cofinal subset of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mmultiscripts><mml:mrow></mml:mrow> <mml:mrow></mml:mrow> <mml:mi>ω</mml:mi></mml:mmultiscripts> <mml:mi>ω</mml:mi> <mml:mo>,</mml:mo> <mml:msup><mml:mo>≤</mml:mo> <mml:mo>∗</mml:mo></mml:msup> <mml:mo>)</mml:mo></mml:mrow> </mml:math> , or equivalently that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi></mml:mrow> </mml:math> ). Here we eliminate that assumption and prove that nonvanishing derived limits, and hence the non-additivity of strong homology, are consistent with any value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>ℵ</mml:mi> <mml:mn>1</mml:mn></mml:msub> <mml:mo>≤</mml:mo> <mml:mi>b</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>d</mml:mi> <mml:mo><</mml:mo> <mml:msub><mml:mi>ℵ</mml:mi> <mml:mi>ω</mml:mi></mml:msub> </mml:mrow> </mml:math> , thus giving a partial answer to a question of Bannister.