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Updated: May 10, 2025

Setting Limits on Supersymmetry Using Simplified Models
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Free subsets in internally approachable models.

P D Welch1

  • 1School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG England.

Archive for Mathematical Logic
|April 28, 2025
PubMed
Summary
This summary is machine-generated.

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This study demonstrates that the Approachable Bounded Subset Property (ABSP) requires large cardinals to establish. It shows that if ABSP holds, an inner model with specific properties exists, directly proving its consistency strength.

Area of Science:

  • Set Theory
  • Large Cardinal Theory

Background:

  • The study investigates the relationship between characteristic functions of internally approachable models and free subsets.
  • It builds upon Pereira's work on the Approachable Free Subsets Property (AFSP) and the related Approachable Bounded Subset Property (ABSP) by Ben-Neria and Adolf.

Purpose of the Study:

  • To determine the large cardinal requirements for establishing the Approachable Bounded Subset Property (ABSP).
  • To directly show that ABSP implies the existence of an inner model with specific properties related to measurables of arbitrarily large Mitchell order.

Main Methods:

  • The research directly proves the implication from ABSP to the existence of an inner model with measurables of arbitrarily large Mitchell order.
  • It avoids the use of PCF (Projective Class Functions) scales, offering a more direct approach compared to prior related work.
Keywords:
Free subsetsInner modelsMeasurabilityPcf theory

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Main Results:

  • The study establishes that the Approachable Bounded Subset Property (ABSP) requires modest large cardinals.
  • A key result is the theorem stating that if ABSP holds for an ascending sequence, then an inner model with measurables of arbitrarily large Mitchell order exists.

Conclusions:

  • The findings provide a direct proof for the consistency strength of ABSP for ascending sequences.
  • This work clarifies the foundational requirements for ABSP, offering a more direct route than previous methods involving PCF scales.