You could define the dimension to be the supremum of the cardinality of all linearly independent sets. In the infinite case this does not trivially give you a

You could define the dimension to be the supremum of the cardinality of all linearly independent sets. In the infinite case this does not trivially give you a basis as 1) the supremum could be strictly larger than the cardinality of all linearly independent sets and 2) adding an extra vector to an infinite linearly independent set doesn't increase it's cardinality, hence there is no reason for the basis to span the entire space. You could also take the infimum of all sets that span the entire space, but you run into similar problems.