Why deos 'rebirth' imply there is no soul?

Buddhists avoid 'reincarnation' instead of 'rebirth'. Especially if we don't mean rebirth into an intermediate state, the term 'reincarnation', to me, seems OK. Why does 'rebirth' imply there is no soul? Is it an etymological thing? Or something about putting the emphasis on 'birth' rather than the body?