Least world records

This morning I saw a tweet from Tony Mann about Berry's Paradox, formulated by Tony as "The smallest integer not identifiable in a tweet". (A paradox, of course, because that phrase identifies a specific integer.)

This reminds me that I have noticed a few times recently people online reciting the joke that goes like: "I hold the world record (jointly) for least world records (this one)". (Here's an arbitrary example found through Twitter search.)

Accept for the moment that such a record exists. Then someone who held one world record for something else (most socks being worn by people spinning plates in a leisure centre, or something) would also jointly hold one world record, so would share the world record for least, so would hold two, so wouldn't share the record...