The fact that one gets contradictions if one assumes that an infinity is given rather than constructing it as a limit is evidence in favor of "infinite set atheism." For example, in Eliezer's example from Jaynes, the real reason for the paradox is that it is completely impossible to pick a random integer from all integers using a uniform distribution: if you pick a random integer, on average lower integers must have a greater probability of being picked.

The most natural thing to conclude is that infinities cannot exist in reality. However, sometimes the most natural conclusion is false, so I'm not sure of this.