Well, modern-day finitists, and through much of history many mathematicians, also thought that there was no such thing as "actual infinity" but only "potential infinity", so that it was illegitimate to think of "the entire collection of natural numbers" as a collected whole. But this isn't what he thinks; he's denied being an actual finitist. (He probably doesn't have the capacity to really understand the distinction between actual and potential infinity). He just thinks there's something special about sets which means they can't be infinite.

My guess is that in some textbook he read or used, it says words to the effect of, "we will only use finite sets" or "the cardinality of a set is the number of elements it contains", and has taken this to be gospel truth. This is, at least, what has happened with order-of-operations, where he misconstrues the statement about simplifying algebraic expressions, "first expand brackets" as being about order-of-operations. So he has form for treating sloppy wording as the word of God and ignoring any other textbook's phrasing.

In the case of order-of-operations I think this is because he believes with religious fervour that mathematics "is never ambiguous" and so the slight ambiguity between the two ways of calculating an expression like a/b(c+d) (namely: do you divide by all of b(c+d) or only by b) must be resolved by some rule, somewhere, and this is where he has found a way.