Richness and Reflection

Just posting regarding an old paper that I think has some relevance: `Richness and Reflection’ contrasts motivations for reflection principles from both a multiversist and universist perspective.

I think this issue merits further examination. More generally, there are many positions between the `extremes’ of a one-universe perspective and an  almost-anything-goes first-order perspective. Justification  of new axioms in set theory is going to have a slightly different flavour for each.

Abstract: A pervasive thought in contemporary philosophy of mathematics is that in order to justify reflection principles, one must hold universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting universism with a Zermelian form of multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, the status of reflection principles as axioms is left open for the Zermelian.

You can find the paper here. If you don’t have access, you can find a pre-print on my academia.edu here.

 

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