Perhaps. But it's simpler than all that. Assume that 70 million disjoint sets all have finite cardinality. Then the cardinality of the union is the sum of the

Perhaps. But it's simpler than all that. Assume that 70 million disjoint sets all have finite cardinality. Then the cardinality of the union is the sum of the cardinalities, and is finite by construction. Contradiction; QED.