Why should I believe Martin's Maximum++?

Question

$\sf MM^{++}$ is a 'good' set-theoretic axiom because it is 'Maximum'. Of course, bigger is better. But I'd like to know exactly how the argument works.

Let me be clear about the question posed:

• What intrinsic and extrinsic justifications do we have to support the claim that $\sf MM^{++}$ is a necessary set-theoretic axiom?

How does $\sf MM^{++}$ clarify the definition of set, what new mathematical results does it bring, and what absoluteness does it bring?

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For those who do not understand the history of Martin's Maximum and the differences between the variants, please read the first 8 pages of https://arxiv.org/abs/1906.10213.

Answer 1

Although it's not really a scholarly publication, the Quanta Magazine article, To Settle Infinity Dispute, a New Law of Logic, gives a good introduction to the topic. That article mentions a conference, Inner Model Theory & Large Cardinals, a 50 year Celebration, but unfortunately I have not been able to find much information about that conference. Another good reference is the 2019 Ph.D. thesis of Jeffrey Schatz, Axiom Selection by Maximization: V = Ultimate L vs Forcing Axioms. In particular, Schatz tries to argue that the "bigger is better" or "maximize" principle favors forcing axioms over V = Ultimate L.