so there isn't a single floating point representation that can represent all of the numbers in [0,1] I agree! What I've done is to propose (an admittedly non-c

so there isn't a single floating point representation that can represent all of the numbers in [0,1] I agree! What I've done is to propose (an admittedly non-computable) set of floating point representations for which you can show a bijection to the real numbers. So if you accept my stretched meaning, which you do so above, then we can answer the title's question with "the cardinality of the real numbers."