Is it possible to randomly pick an integer from an infinite set of integers?

I was disputing a friend’s hypothetical about an infinite lottery. They insisted you could randomly pick 6 integers from an infinite set of integers and each integer would have a zero chance of being picked. I think you couldn’t have that, because the probability would be 1/infinity to pick any integer and that isn’t a defined number as far as I know. But I don’t know enough about probability to feel secure in this answer.