elementary set theory – Can we use up every whole number in the set of Natural numbers?

In this video, Vsauce says “We’ve used up every single whole number, the entire infinity of them…”

Then a few seconds later he says the opposite, “You will never run out of members of either set…”

Could Vsauce have used better wording to avoid sounding like he was putting forward a contradiction? Or is this an actual weakness in the argument, i.e. we are tacitly assuming that it is possible to exhaust an endless set?