Uh, uh, uh, I mean, the complete mathematical structure should be something like this.Let me give you an example.
Icarus cardinality: there exists an L (V_ λ+ 1, lcuras) non trivial basic embedding, with a critical point below λ, Icarus exists in V_ λ+ 2-L (V_ λ+ 1) .
Integrity axiom:
I3: Presence of V λ Embedding into its own non trivial basic.
I2: V has a non-trivial fundamental embedded into the containing V λ The transitive class M of, λ Is the first fixed point above the critical point.
I1: V λ+ 1 to its own non trivial basic embedding.
I0: Presence of L (V λ+ The non trivial basic embedding of 1), its critical point< λ Axiom.
Super Reinhardt cardinality: Super Reinhardt cardinality for any ordinal number α, There is a j: V → V with j (K)> α And it has a critical point K, which can be called 0=1 because a sufficiently large cardinality axiom can lead to inconsistency, making all propositions in the system true.
Berkeley Club: Base κ Is the Berkeley base, if for any band κ The transitive set k ∈ M of and any ordinal number α<κ, There will always be an elementary embedding j: M<M and crit j<k. If there is indeed a Berkeley cardinality, then there will be a forced expansion absolute, which makes the smallest Berkeley cardinality co tailed ω, By comparing κ Applying certain conditions seems to enhance the Berkeley property, if κ It's Berkeley and α,α∈ M and M have transitivity, then for any α< k. There is a j: M<M and α< Crit j<k and crit j (a)=a, for any transitive M ∋ k, there exists j: M ≺ M and crit j<K, with a cardinality of Berkeley, and only for any transitive set M ∋ k κ Existence of j: M ≺ M and α< Crit j<k, therefore δ≥ K, δ Also known as Berkeley, the smallest Berkeley base is also known as δ_α, call κ For club Berkeley, if κ It is regular and for all clubs → C ⊆ κ And all bands κ The transitive set M ∈ M of; Have j ∈ ε (M) And crit (j) ∈ C, denoted as κ For limit club Berkeley, it is a club Berkeley base/limit Berkeley base. If K is the smallest Berkeley, then y<k.