You can make an exception for a particular bit pattern for an irrational, like Pi, just like you can use a particular bit pattern to represent -0 and NaN. Since the title doesn't specify a floating point representation, I thereby invoke all possible floating point representations. I construct a set of such representations with the cardinality of Reals like so: For any given real number X, just formulate a representation like single-precision IEEE 754 floating-point but which substitutes X for NaN.

The article doesn't fall into this trap, but the title does.

Now, if you think your objection sinks my definition of "floating point number," note that it also sinks IEEE 754 floating-point.

EDIT: So you formulate a particular definition of "floating point number" for which you are right. To which I say: So What?