In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992.
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A Berkeley cardinal is a cardinal κ in a model of Zermelo–Fraenkel set theory with the property that for every transitive set M that includes κ and α < κ, there is a nontrivial elementary embedding of M into M with α < critical point < κ. [1] Berkeley cardinals are a strictly stronger cardinal axiom than Reinhardt cardinals, implying that they are not compatible with the axiom of choice.
A weakening of being a Berkeley cardinal is that for every binary relation R on Vκ, there is a nontrivial elementary embedding of (Vκ, R) into itself. This implies that we have elementary
- j1, j2, j3, ...
- j1: (Vκ, ∈) → (Vκ, ∈),
- j2: (Vκ, ∈, j1) → (Vκ, ∈, j1),
- j3: (Vκ, ∈, j1, j2) → (Vκ, ∈, j1, j2),
and so on. This can be continued any finite number of times, and to the extent that the model has dependent choice, transfinitely. Thus, plausibly, this notion can be strengthened simply by asserting more dependent choice.
While all these notions are incompatible with Zermelo–Fraenkel set theory (ZFC), their consequences do not appear to be false. There is no known inconsistency with ZFC in asserting that, for example:
For every ordinal λ, there is a transitive model of ZF + Berkeley cardinal that is closed under λ sequences.
While lesser known, there is a SuperBerkeley ordinal which is defined informally as:
A SuperBerkeley cardinal is a cardinal number called h that satisfies two conditions:
1. There are Berkeley cardinals below h, and they are unbounded below h. This means: no matter how far below h you look, you can always find another Berkeley cardinal. So h is a limit of Berkeley cardinals.
2. h itself has the Berkeley property. This means the following:
Whenever you take any well‑behaved collection of sets that contains h, and whenever you choose any smaller number below h, there is a nontrivial self‑map of that collection that:
preserves all truths about the collection,
actually moves some elements (so it’s not the identity), and
has its “first moved point” somewhere between your chosen number and h.