Large cardinals as principles of structural reflection. (English) Zbl 07665537
Summary: After discussing the limitations inherent to all set-theoretic reflection principles akin to those studied by A. Lévy et. al. in the 1960s, we introduce new principles of reflection based on the general notion of Structural Reflection and argue that they are in strong agreement with the conception of reflection implicit in Cantor’s original idea of the unknowability of the Absolute, which was subsequently developed in the works of Ackermann, Lévy, Gödel, Reinhardt, and others. We then present a comprehensive survey of results showing that different forms of the new principle of Structural Reflection are equivalent to well-known large cardinal axioms covering all regions of the large-cardinal hierarchy, thereby justifying the naturalness of the latter.
MSC:
| 03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |
| 03E55 | Large cardinals |
| 03E65 | Other set-theoretic hypotheses and axioms |
| 03Exx | Set theory |
| 00A30 | Philosophy of mathematics |
| 03A05 | Philosophical and critical aspects of logic and foundations |
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