Full Idea
It must not be supposed that we can obtain a new transfinite cardinal by merely adding one to it, or even by adding any finite number, or aleph-0. On the contrary, such puny weapons cannot disturb the transfinite cardinals.
Gist of Idea
You can't get a new transfinite cardinal from an old one just by adding finite numbers to it
Source
Bertrand Russell (The Principles of Mathematics [1903], §288)
Book Reference
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.311
A Reaction
If you add one, the original cardinal would be a subset of the new one, and infinite numbers have their subsets equal to the whole, so you have gone nowhere. You begin to wonder whether transfinite cardinals are numbers at all.