Infinite set atheists doubt that infinite sets really exist. That is, they believe that no collection of existing things contains an infinite quantity of elements. An infinite set atheists may further believe that the concept of an infinite quantity is unnecessary or even incoherent. This position holds that you shouldn't need to use infinite quantities even when you consider a collection of possible things. No one has demonstrated an incoherence in the modern mathematical concept of infinite quantities. However, no one has demonstrated that no such incoherence exists.
Part of the motivation for infinite set atheism is that very strange things happen when we deal with infinite quantities in mathematics. Untutored intuition wants to say that the quantity of natural numbers is larger than the quantity of even natural numbers. However, this turns out not to be the case. Two sets contain the same quantity when we can put them in one-to-one correspondence with each other: match each element of one set with one unique element in the other set so that no element in either set is left unmatched. For example, we can showing that {1, 2, 3} and {4, 5, 6} contain the same quantity of elements by pairing 1 with 4, 2 with 5, and 3 with 6, which covers all the elements. If a set is finite, then removing any element from it will produce a set containing a smaller quantity of elements. Infinite sets, however, behave fundamentally differently: the infinite set of all natural numbers can be put into one-to-one correspondence with the infinite set of all even numbers with the correspondence n <--> 2n: pair 1 with 2, 2 with 4, 3 with 6, and so on for each natural number n.