Benedikt Löwe (University of Amsterdam)

Inaccessible cardinals

The tutorial will assume some knowledge of naive set theory and start by reviewing the basics of the theory of ordinals and cardinals, in particular regular cardinals. After observing that all successor cardinals are regular and that there are non-regular limit cardinals, one natural and fundamental question of this theory is "Are there any regular limit cardinals?". This innocent question turns out to be more complicated than it seems at first sight.
    We will show that you cannot prove (in ZFC) that there are regular limit cardinals (called "inaccessible cardinals"), and that if they exist, they must be outrageously large. So, why care? In the second half of the talk, we shall see how the existence of some inaccessible cardinals relates to fundamental questions about set theory of the real number line such as Lebesgue measurability.

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