"Assuming the continuum hypothesis for the sake of this notation" is being deliberately obfuscatory, encouraging confusion, and requiring other mathematicians t

"Assuming the continuum hypothesis for the sake of this notation" is being deliberately obfuscatory, encouraging confusion, and requiring other mathematicians to do extra work to translate your results to the more general context. There's perfectly good notation for the bet series, and no need to use aleph as a substitute when that simply isn't what it means.