Antonyms for the phrase "set heart on" would include "disinterest," "detachment," and "indifference." These words suggest a lack of emotional investment or passion towards a particular goal or desire. Other antonyms could include "lose sight of," "abandon," or "neglect," conveying a sense of giving up on a previously held ambition or objective. In contrast, synonyms for "set heart on" might include "aspire to," "dream of," or "aim for," indicating a strong sense of drive and determination to achieve something significant. Ultimately, the choice of antonym will depend on the particular context and tone of the writing.