Infallibility

Infallibility refers to an inability to be wrong. It can be applied within a specific domain, or it can be used as a more general adjective. The term has significance in both epistemology and theology, and its meaning and significance in both fields is the subject of continued debate.
In philosophy.
Epistemology, a branch of philosophy, is concerned with the question of what, if anything, humans can know. The answer to the issue of whether or not a human can be infallible depends on the philosophical school.
In theology.
Christianity.
"The Oxford Dictionary of the Christian Church" defines infallibility as "Inability to err in teaching revealed truth". Catholic and Eastern Orthodox theology claim that the Church is infallible, but disagree as to where infallibility exists, whether in doctrines, scripture, or church authorities.
In Catholic theology, Jesus, who is the Truth, is infallible, but only a special act of teaching by the church's bishops may properly be called "infallible". According to the First Vatican Council (1869–71) and as reaffirmed at Vatican II (1962–1965), the earthly head of the Catholic Church, the Pope, is infallible when speaking "ex cathedra" on matters of faith and morals (that is, when he explicitly intends to use his papal office to teach the whole Church definitively and irreformably on matters which deal directly with faith and morals). However, papal infallibility does not extend beyond such cases, thus making it possible for a Pope to sin and to be incorrect. Papal infallibility also belongs to the body of bishops as a whole, when, in doctrinal unity with the pope, they solemnly teach a doctrine as true.
In contrast, Protestant and non-denominational Christian churches believe that the Christian Church is indeed fallible, as evidenced by the requirement of Christ's sacrifice on the cross to pay for the sins of the world, including those of his Church, and that only God's word in scripture is infallible. They also completely reject the Catholic claim regarding papal infallibility, citing not just scriptural reasons, but also the many times popes have contradicted each other and the history of mistakes committed by many popes throughout Roman Catholic Church history.
Because of the complexity in defining infallibility, some Protestant and non-denominational views confuse papal infallibility with impeccability, as if the Pope were immune from sin. This is not the Catholic Church's doctrine, which concedes that Popes can sin and may even contradict one another's personal theological opinions. It is evident that the Pope can sin (which every Pope explicitly states in the Confiteor, and whenever he receives the Sacrament of Penance) and a Pope may even succumb to heresy, as suggested in the condemnation by the Third Council of Constantinople (13th and 16th session) which anathematized Pope Honorius I for supporting the heresy of Monothelitism, and which Pope John XXII admitted when he retracted his views on the Beatific vision.
Islam.
Universal teachings.
Islam teaches that the teachings and guidance by the Prophets with regard to bringing the message of the One true God is infallible. Islam also teaches that the Quran is an infallible text.
Additional Shi'a teachings.
In Shi'a theology, the belief is that the Ahl al-Bayt, including Muhammad, his daughter Fatima Zahra and Shi'a Imams are all infallible. It is believed that they are infallible in the sense that all statements or teachings made by them can be relied on to be certainly true, that all information believed by themselves is true, and that they have complete knowledge about right and wrong and never intend to disobey God. It is also held by Shi'as that there were 124,000 Prophets, beginning with Adam and ending with Muhammad - with all, including the last, being infallible in the same sense as the Ahl al-Bayt.
Judaism.
The notion of infallibility in Judaism as it relates to the Tannaim and Amoraim of the Talmud, as well as the Rishonim, Achronim and modern day Gedolim is one surrounded by debate.
Some who reject infallibility cite the Talmud, Pesachim 94b:
The words of the Mishnah are commented on by numerous commentators, and Judah Halevi argues that evidence mounts that the Geonim and Maimonides perceived that the sages of the Talmud "erred in a matter of astronomy". Maimonides wrote that the great sages are not expected to advocate positions perfectly in-line with modern science because they were "scholars of that generation," often basing their assessments of what "they learned from the scholars of the era."
In the Hasidic tradition, however, infallibility is taught in the Chabad tradition in connection with a rebbe.
Vaishnavism (Hinduism).
In the Bhagavad Gita, Krishna explains to Arjuna (Bg 15.16 to 15.20)