since on⊂l⊆v, properties of ordinals that depend on the absence of a function or other structure (i.e. π formulas) are preserved when going down from v to l. hence initial ordinals of cardinals remain initial in l. regular ordinals remain regular in l. weak limit cardinals become strong limit cardinals in l because the generalized continuum hypothesis holds in l. weakly inaccessible cardinals become strongly inaccessible. weakly mahlo cardinals become strongly mahlo. and more generally, any large cardinal property weaker than 0 (see the list of large cardinal properties) will be retained in l.