Berkeley cardinal

In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar in about 1992.

A Berkeley cardinal is a cardinal κ in a model of ZF with the property that for every transitive set M that includes κ, there is a nontrivial elementary embedding of M into M with critical point below κ. 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 V_κ, there is a nontrivial elementary embedding of (V_κ, R) into itself. This implies that we have elementary

j₁, j₂, j₃, ...

j₁: (V_κ, ∈) → (V_κ, ∈),

j₂: (V_κ, ∈, j₁) → (V_κ, ∈, j₁),

j₃: (V_κ, ∈, j₁, j₂) → (V_κ, ∈, 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 ZFC, their $\Pi _{2}^{V}$ 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.

References

Chen, Evan; Koellner, Peter (2015), Math 145b Lecture Notes (PDF)
Koellner, Peter (2014), The Search for Deep Inconsistency (PDF)

External links

Berkeley cardinals in Cantor's attic

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