  The large cardinals considered thus far (including κ(ω)) are known as small large cardinals.  A large cardinal is small if the associated large cardinal axiom can hold in Gödel's constructible universe L, that is, if   “V ⊨ κ is a φ-cardinal” is consistent, then   “L ⊨ κ is a φ-cardinal” is consistent.  Otherwise the large cardinal is large.