In ZF it is possible to have an infinite cardinal which is not a well-ordered cardinal. It is the cardinal successor of aleph 0 . Every successor cardinal is a well-ordered cardinal. Every well-ordered cardinal that is not a successor cardinal is a limit cardinal. Any cardinal of the form aleph a , where a is a limit ordinal, is a limit cardinal.
Some authors require limit cardinals to be uncountable.
The reason that I prefer the definition I give is that it makes the partitions of cardinals much nicer, since we do not need to consider 0 and aleph 0 special cases, i.e., well-ordered cardinals which are neither successors nor limits. Also, with the definition I use aleph 0 is considered a regular limit cardinal and a regular strong limit cardinal. These special properties of aleph 0 are what inspired the definition of weakly inaccessible and inaccessible cardinals. Every strong limit cardinal is a limit cardinal. Another way to say this is to say that it is not possible to represent k as the supremum of fewer than k smaller ordinals, or k cannot be written as the sum of fewer smaller cardinals.