No, I don't think there's very much room for controversy here. I mean, I don't know what exactly Hotz have said, since there was no quote (and, honestly, I'm not that interested either), but if somebody is simply saying "the cardinality of integers is the same as real numbers" and leaving it at that, he is just plainly wrong.
Math is all about definitions and what follows from these definitions. So, you can define "integer", "real number", "cardinality", "equals" and so on however you like, and make all sorts of correct statements — as everyone will see by following your arguments all way from the definition/axiom and until the very end of your proof. But if you don't provide any definitions of your own, then you rely on some other definitions, and everyone has no other choice than assume that these are the very much "mainstream" ones, as you are referring to them as if they are well-known.
Now, it is unquestionably true (and easily provable) that the set of all computable numbers is countable, and anyone who says otherwise is wrong. But unless you specifically define real numbers as a subset of computable numbers, as constructivists are inclined to do, your listeners won't assume that, since this is not how real numbers are generally defined, and by virtue of not providing your own definition you are implicitly referring to a "general definition". (And, honestly, you shouldn't even call any subset of computable numbers "a set of Real numbers": this name is already taken.)
These general definitions and assumptions lead to all sorts of complications, and I personally have my doubts that real numbers exist in any meaningful sense (although I'm not committed to that statement, since there are several mathematical constructs that I would like to dismiss as "clearly nonsense", except they allow us to prove some very "no-nonsense" stuff — I don't know how to deal with that, and I never heard that anybody does). But I definitely cannot say that cardinality of integers is the same as the cardinality of reals, because this is simply not true under the common definitions (which is easily provable). (And less importantly, but worth saying that the contrary is not proven by constructivist methods — as half of the actually useful math in general ends up being, unfortunately).
So, as a somebody, who doesn't quite believe in non-computable numbers, I am very sympathetic to anybody who says that Real numbers do not exist. I don't understand how could they, what does it mean for an object that we cannot define to exist. Yet, I can accept (as a game) some well-known theory which talks about these non-quite-existant "Real numbers", and prove some statements about them, and one of these easily provable statements is that cardinality of continuum does not equal the cardinality of Natural numbers.