When did Christ claim the apostles (and their sucessors) would be infailable?
When did the concept of infailablity begin?
-Pax Vobiscum
From the beginning. It seems you are asking about when the infallibility of the Church was
defined. Infallibility of the Church in scriptures:
The Church is Infallible and Supernatural
Isa. 35:8, 54:13-17 - this prophecy refers to the Church as the Holy Way
where sons will be taught by God and they will not err. The Church has been given the gift of infallibility when teaching about faith and morals, where her sons are taught directly by God and will not err. This gift of infallibility means that the Church is prevented from teaching error by the power of the Holy Spirit (it does not mean that Church leaders do not sin!)
Acts 9:2; 22:4; 24:14,22 - the early Church is identified as the “Way” prophesied in Isaiah 35:8
where fools will not err therein.
Matt. 10:20; Luke 12:12 - Jesus tells His apostles it is not they who speak, but the Spirit of their Father speaking through them.
If the Spirit is the one speaking and leading the Church, the Church cannot err on matters of faith and morals.
Matt. 16:18 - Jesus promises the gates of Hades would never prevail against the Church. This requires that the Church teach infallibly.
If the Church did not have the gift of infallibility, the gates of Hades and error would prevail. Also, since the Catholic Church was the only Church that existed up until the Reformation, those who follow the Protestant reformers call Christ a liar by saying that Hades did prevail.
Matt. 16:19 - for Jesus to give Peter and the apostles, mere human beings,
the authority to bind in heaven what they bound on earth requires infallibility. This is a gift of the Holy Spirit and has nothing to do with the holiness of the person receiving the gift.
Matt. 18:17-18 -
the Church (not Scripture) is the final authority on questions of the faith. This demands infallibility when teaching the faith. She must be prevented from teaching error in order to lead her members to the fullness of salvation.
Matt. 28:20 - Jesus promises that He will be with the Church always.
Jesus’ presence in the Church assures infallible teaching on faith and morals. With Jesus present, we can never be deceived.
Mark 8:33 - non-Catholics sometimes use this verse to down play Peter’s authority. This does not make sense. In this verse, Jesus rebukes Peter to show the import of His Messianic role as the Savior of humanity. Moreover, at this point, Peter was not yet the Pope with the keys, and Jesus did not rebuke Peter for his teaching. Jesus rebuked Peter for his lack of understanding.
Luke 10:16 - whoever hears you, hears me. Whoever rejects you, rejects me.
Jesus is very clear that the bishops of the Church speak with Christ’s infallible authority.