Free subsets in internally approachable models

We consider a question of Pereira as to whether the characteristic function of an internally approachable model can lead to free subsets for functions of the model. Pereira isolated the pertinent <i>Approachable Free Subsets Property</i> (AFSP) in his work on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>pcf</mml:mtext></mml:math> -conjecture. A recent related property is the <i>Approachable Bounded Subset Property</i> (ABSP) of Ben-Neria and Adolf, and we here directly show it requires modest large cardinals to establish: <b>Theorem</b> <i>If ABSP</i> holds for an ascending sequence <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub><mml:mrow><mml:mo>⟨</mml:mo> <mml:msub><mml:mi>ℵ</mml:mi> <mml:msub><mml:mi>n</mml:mi> <mml:mi>m</mml:mi></mml:msub> </mml:msub> <mml:mo>⟩</mml:mo></mml:mrow> <mml:mi>m</mml:mi></mml:msub> </mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>n</mml:mi> <mml:mi>m</mml:mi></mml:msub> <mml:mo>∈</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> then there is an inner model with measurables <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>κ</mml:mi> <mml:mo><</mml:mo> <mml:msub><mml:mi>ℵ</mml:mi> <mml:mi>ω</mml:mi></mml:msub> </mml:mrow> </mml:math> of arbitrarily large Mitchell order below <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℵ</mml:mi> <mml:mi>ω</mml:mi></mml:msub> </mml:math> , that is: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>sup</mml:mo> <mml:mfenced><mml:mi>α</mml:mi> <mml:mo>∣</mml:mo> <mml:mo>∃</mml:mo> <mml:mi>κ</mml:mi> <mml:mo><</mml:mo> <mml:msub><mml:mi>ℵ</mml:mi> <mml:mi>ω</mml:mi></mml:msub> <mml:mi>o</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>κ</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>≥</mml:mo> <mml:mi>α</mml:mi></mml:mfenced> <mml:mo>=</mml:mo> <mml:msub><mml:mi>ℵ</mml:mi> <mml:mi>ω</mml:mi></mml:msub> </mml:mrow> </mml:math> . A result of Adolf and Ben Neria then shows that this conclusion is in fact the exact consistency strength of ABSP for such an ascending sequence. Their result went via the consistency of the non-existence of continuous tree-like scales; the result of this paper is direct and avoids the use of PCF scales.