It sounds funny to ever say "the AoC is wrong," though, because axioms are things you get to choose as true or false. They are essentially subjective, and in practice are as "socially true" as they are commonly used. I guess the author is arguing that the more intuitively satisfying set of axioms excludes AoC.
I personally disagree, although I admit there's still room for debate because of compelling arguments on either side (most of which are appeals to intuition by the nature of the problem).
Here's a small counter-argument: Since we're appealing to intuition, consider that we're asking a countably infinite number of people to (a) see and reason based on seeing an infinite sequence in front of them; and (b) all memorize a specific function which is essentially from all the reals in [0,1] to a subset of [0,1], which we may think of as binary numbers with infinite expansions. (The intuitive point is clear without having to worry about 0.01111... being the same value as 0.1). This function is insanely weird and hard to describe succinctly; basically, it must be memorized. My intuition tells me that as soon as we expect agents to perform these feats, they are no longer humans and no longer Turing machines. Since this memorized function is so weird, they are even beyond some kind of infinite-time Turing machines with finite memories (or infinite-memory TMs with finite time). I think they are even beyond infinite-time Turing machines with countably-infinite memories, although I haven't proven this carefully.
In the long run, I suspect an alternative axiomitization (or possibly an elegant classification of set theory statements) will shed light on these problems, somewhat analogous to the way we now consider non-Euclidean geometry as interesting and "true" in its own universe that we mentally separate from the Euclidean plane.
[ps Not all comments about axioms are appeals to intuition. E.g., you can prove some are independent or dependent. But arguments along the lines of this post are appeals to intuition when used for/against AoC.]