I apologize for stating that incorrectly. We're talking about calculations with infinity, that is what do you get when you ask for a subset of infinity, such as all odd numbers or, in this case, half. The resulting set of numbers will always be infinite.
{1, 2, 3} is a finite subset of the infinite series of integers.
What he should have said is that "infinite" can be defined as the size of any set that can be put into a one-to-one correspondence with some (not every) proper subset of itself.
Thus, the set of all real numbers can be put into 1-to-1 correspondence with the set of all real numbers between 0 and 1. The set of all rational numbers can be put into a 1-to-1 correspondence with the set of all integers. The set of all integers can be put into a 1-to-1 correspondence with the set of all positive integers. And so on.
Obviously the notion of "infinite quantities" is an entirely different animal from the notion of "quantities" that we actually observe in the real world. Many people seem not to realize this and, despite the conceptual impossibility, attempt in language to concretize "infinite quantities" as some extension of observable quantities.