"Infinity" may not be a number, but omega is a number, and it is transfinite. Omega is the first transfinite ordinal. You are certainly correct, however, that

"Infinity" may not be a number, but omega is a number, and it is transfinite. Omega is the first transfinite ordinal. You are certainly correct, however, that Cantor's hotel does not need to have any rooms with transfinite room numbers.