Are the Gospels Inerrant?

Unfortunately for that contention there are several problems...1st - Augustus only conducted censuses in 12 BCE, 8 BCE, 6 CE and 14 CE, this means that if Jesus was born during either of the 1st century BCE, he would have started his ministry and been executed long before John the Baptist or Pilate were on the scene, which would negate the historic value all the gospels. 2nd - We have an accurate listing of the governors of Syria from the inception of the province to the fall of the Byzantine Empire (the successor of the Roman Empire) and during the 1st century BCE Cyrenius is not mentioned as governor. 3rd - there were no dual governancies, anyone with any knowledge of Roman culture would understand that no one of consular rank would betray his "dignitas" (standing in society) by subjugating himself to another of equal rank - imperium was never shared. 4th - we can track Cyrenius career and at the time of both of the 1st century BCE censuses, he was in Rome. 5th - Galilee was an independent nation at the time of all the censuses, thus Joseph and Mary would have no reason to respond to the dictates of an alien ruler. 6th - only the 6 CE census involved anyone other than Roman citizens...so no, I don't think there was any possibility of any other census...Apologists have tried many ploys to make another census to bring Matthew and Luke into line but none work - face it, the NT is mythology based on mythology based on mythology and the religion is no more valid than Buddhism, Mithran, Zoroasterism, Judaism or Taoism. :patriot:

This date neatly coincides with another fact that Luke recorded - the time that Cyrenius was governor of Syria. Therefore, Luke reports that Jesus was born in 6 CE. This is a 14 to 12 year difference between the two dates offered by the gospels and since only these two gospels mention anything about the birth and youth of Jesus, they are the only source we have.

the census of 1bc/ad was not an ordinary census, but rather one that was ordered by the Roman Senate as part of an empire-wide celebration of the peace and prosperity that Augustus had brought to the Empire following the assassination of Caesar and the post-assassination civil war.

but there is only one piece of non-Biblical documentation to indicate that Cyrenius was governor more than once.

Why should both Gospels have to give the exact same details for either to be accepted as accurate?

What difference does it make why Mary and Joseph were in Bethlehem when Jesus was born?

Your objection to the Gospels on historical grounds is based on assumptions about details that are not really discussed in the Bible.