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Instances of papal infallibility
Many non-Catholics, and even some Catholics, wrongly believe that the doctrine teaches that the Pope is infallible in everything he says. In reality, the use of papal infallibility is quite rare.
Nearly all Catholic theologians agree that both Pope Pius IX's 1854 definition of the dogma of the Immaculate Conception of Mary, and Pope Pius XII's 1950 definition of the dogma of the Assumption of Mary are instances of papal infallibility. However, theologians disagree about what other documents qualify.
After Pope John Paul II's apostolic letter Ordinatio Sacerdotalis (On Reserving Priestly Ordination to Men Alone) was released in 1994, a few commentators speculated that this might be an exercise of papal infallibility (for an example, see [1]). The Congregation for the Doctrine of the Faith (CDF) stated that it was not. However, the CDF stated that the content of this letter had already been taught infallibly by the ordinary and universal magisterium.
The Vatican itself has given no list of papal statements considered to be infallible. A 1998 commentary by Cardinal Ratzinger and Cardinal Bertone, the leaders of the CDF, listed a number of instances of infallible pronouncements by popes and by ecumenical councils, but explicitly stated that this was not meant to be a complete list.
The number of infallible pronouncements by ecumenical councils is significantly greater than the number of infallible pronouncements by popes.