Please read the narration which proved that the Prophet(pbuh) did not appoint Abu Bakr as caliph. This narration also proves that Abu Bakr is the one who made Umar bin Khattab as Caliph after him.
http://www.sunnah.com/muslim/33/13
We want to comment more on this issue.
We see some more interesting things. When Abu Bakar was close to dying, he appointed Omar bin khittab (the one who worked hard to give government to Abu Bakar).
Why not Abu Bakar acted upon the Sunnah of Prophet Mohammad? Did Abu Bakar have more vision than Prophet Mohammad that he appointed Omar bin Khattab after him?
Another interesting thing: when Omar bin Khattab was near to death, his son Abdullah bin Omar came to him and told him if somebody has goats and sheeps, he makes the arrangement for them. He asked him to make some arrangement for the people who will be in-charge after him. Omar replied that I will make an arrangement. Omar bin Khattab made 6 members committee and made one person head to choose the caliph.
We can also see that Omar bin Khattab did not act on the Sunnah of Prophet, Was Omar bin Khattab smarter than holy Prophet Mohammad (Astaghferullah)!
(As Sunnis say that the Prophet did not make anybody Caliph. It means Abu Bakar and Omar both did not follow the Prophet's way)
Our Sunni brothers propagates that the Caliphate of these four caliphs is symbol for Muslim and we should impose it to the whole world.
The question is what is the foundation of this Caliphate? What is the principle for this Caliphate?
Is this based on appointment by the holy prophet? Did the prophet appointed those caliphs? The answer is no.
Is this based on selection committee? The answer is no, the prophet did not make any committee to choose the caliphs.
Is this based on public election? The answer is no. Only some people gathered in Saqifa, left the holy prophet's body un-buried. They divided this power between them.
Most of the companions of the Prophet were not there; the Banu Hashim family of the Prophet, Imam Ali and other prominent personalities were not there.
All other Muslim tribes living outside of Medina and other areas did not know anything about it, they were just forced to accept it and follow it. This was not based on public election.
As we have mentioned that second and third caliphs were made with different ways. There is no criterion and no principal for us to say that this Caliphate is based on those foundations.