Did the Prophet (peace be upon him) not appoint his successor after returning from the Farewell Pilgrimage?

The Prophet, peace and blessings be upon him, did not appoint a successor after him. Rather, he left the matter to be decided by consultation among the Muslims. He had put Abu Bakr forward to lead the people in prayer during his illness, and so the Muslims chose him for their caliphate. The Prophet's sermon at Ghadir Khumm is preserved, where he said: "I am leaving among you two weighty things: the first of them is the Book of Allah, and the second is my family (Ahl al-Bayt)," then he reiterated the reminder about his family. His family (Ahl al-Bayt) are the family of Ali, the family of Aqil, the family of Ja'far, and the family of Abbas, upon whom sadaqa (charity) is forbidden. Thus, it is not proven that the Prophet, peace and blessings be upon him, designated a caliph or explicitly bequeathed the caliphate to a specific individual.