Richness and Reflection

Philosophia Mathematica 24 (3):330-359 (2016)


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.


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Neil Barton
Universität Konstanz

References found in this work

Realism in Mathematics.Penelope Maddy - 1990 - Oxford, England and New York, NY, USA: Oxford University Prress.
Foundations of Set Theory.Abraham Adolf Fraenkel & Yehoshua Bar-Hillel - 1958 - Atlantic Highlands, NJ, USA: North-Holland.
The Potential Hierarchy of Sets.Øystein Linnebo - 2013 - Review of Symbolic Logic 6 (2):205-228.
What is Cantor's Continuum Problem?Kurt Gödel - 1947 - In Solomon Feferman, John Dawson & Stephen Kleene (eds.), Journal of Symbolic Logic. Oxford University Press. pp. 176--187.

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