Thus, for m > 1, every cardinal that is either Π-indescribable or Σ-indescribable is both Π-indescribable and Σ-indescribable and the set of such cardinals below it is stationary.

The consistency strength is Σ-indescribable cardinals is below that of Π-indescribable, but for m > 1 it is consistent with ZFC that the least Σ-indescribable exists and is above the least Π-indescribable cardinal ( this is proved from consistency of ZFC with Π-indescribable cardinal and a Σ-indescribable cardinal above it ).

Measurable cardinals are Π-indescribable, but the smallest measurable cardinal is not Σ-indescribable.

If α is infinite then α-indescribable ordinals are totally indescribable, and if α is finite they are the same as Π-indescribable ordinals.

A cardinal number κ is called Π-indescribable if for every Π < sub > m </ sub > proposition φ, and set A ⊆ V < sub > κ </ sub > with ( V < sub > κ + n </ sub >, ∈, A ) ⊧ φ there exists an α < κ with ( V < sub > α + n </ sub >, ∈, A ∩ V < sub > α </ sub >) ⊧ φ.

The cardinal number κ is called totally indescribable if it is Π-indescribable for all positive integers m and n.

A cardinal is inaccessible if and only if it is Π-indescribable for all positive integers n, equivalently iff it is Π-indescribable, equivalently iff it is Σ-indescribable.

A cardinal is Σ-indescribable iff it is Π-indescribable.

The property of being Π-indescribable is Π.

For m > 1, the property of being Π-indescribable is Σ and the property of being Σ-indescribable is Π.

It is delivered only in a secret consistory at which the cardinals alone are present.

Most cardinals are already bishops at the time of their appointment, the majority being archbishops of important archdioceses or patriarchs, and a substantial portion of the rest already titular archbishops serving in the Vatican.

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality ( size ) of sets.

This sequence starts with the natural numbers including zero ( finite cardinals ), which are followed by the aleph numbers ( infinite cardinals of well-ordered sets ).

If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.

Pope Paul VI also increased the number of cardinal bishops by giving that rank to patriarchs of the Eastern Catholic Churches who are made cardinals.

Cardinal bishops ( cardinals of the Episcopal Order ) are among the most senior prelates of the Catholic Church.

Though nowadays most cardinals are also bishops, the term Cardinal bishops only refers to the cardinals that are titular bishops of one of the " suburbicarian " sees.

The Latin Rite Patriarchs of Lisbon and Venice, while in practice always made cardinals at the consistory after they take possession of their sees, are made cardinal priests, not cardinal bishops.

Cardinal priests are the most numerous of the three orders of cardinals in the Catholic Church, ranking above the cardinal deacons and below the cardinal bishops.

Pope Paul VI abolished all administrative rights cardinals had with regard to their titular churches, though the cardinal's name and coat of arms are still posted in the church, and they are expected to preach there if convenient when they are in Rome.

He had certain ceremonial duties in the conclave that have effectively ceased because he would generally have already reached age 80, at which cardinals are barred from the conclave.

The cardinal deacons are the lowest-ranking cardinals.

For those priests over 80 who became cardinal-deacons and were not ordained to the episcopacy, this is the highest position they can normally attain in the Church hierarchy ( though all cardinals rank above bishops in rank and order of precedence, those cardinals who are not bishops do not have the right to perform the functions reserved solely to bishops, such as ordination ).

With the revision of the Code of Canon Law promulgated in 1917 by Pope Benedict XV, only those who are already priests or bishops may be appointed cardinals.

Today, cardinals are named in pectore to protect them or their congregations from reprisals if their identities were known.

Where it differs from a " true " axiomatic set theory book is its character: there are no long-winded discussions of axiomatic minutiae, and there is next to nothing about advanced topics like large cardinals.

Under present canon law, the pope is elected by the cardinal electors, comprising those cardinals who are under the age of 80.

It is because of the scarlet color of cardinals ' vesture that the bird of the same name has become known as such.

At the same time, the excluded cardinals, most of whom were supporters of the Pierleoni family, elected Pietro Pierleoni, who took the name Anacletus II, throwing the church once again into schism.

Unable any longer to resist the urgency of the French cardinals, and with numerous cities of the Papal States in revolt, Urban boarded a ship at Corneto heading for France on 5 September 1370, arriving back at Avignon on the 24th of the same month.

He lived very parsimoniously and exhorted the cardinals to do the same.

" Later, on the same day, Hildebrand was conducted to the church of San Pietro in Vincoli and elected Pope there in legal form by the assembled cardinals, with the due consent of the Roman clergy, amid the repeated acclamations of the people.

In a sign of his increasing secretiveness and paranoia, he added two more cardinals secretly at the same consistory, and four more at the beginning of 1471, expecting to reveal them only in his testament.

After the death of Honorius II, Petrus Leonis, under the name of Anacletus II, was elected as Pope by a majority of the cardinals and with the support of the people of Rome on the same day as a minority elected Innocent II.

The former of these, after premising " that the Pope of Rome, accroaching to him the seignories of possession and benefices of the holy Church of the realm of England doth give and grant the same benefices to aliens which did never dwell in England, and to cardinals, which might not dwell here, and to others as well aliens as denizens, as if he had been patron or advowee of the said dignities and benefices, as he was not of right by the laws of England ..." ordained the free election of all dignities and benefices elective in the manner as they were granted by the king ’ s progenitors.

In a letter of the same date addressed by the cardinals at Avignon to the entire hierarchy of Sicily, special stress was laid on the fact that the rebellious fugitives had elected a superior general, provincials, and guardians.

Thus the 1969 Instruction states that, for cardinals and bishops, " the elbow-length cape, trimmed in the same manner as this cassock, may be worn over it ".

Pius XII spoke on the same topic to a consistory of cardinals, in his Christmas messages and to numerous academic and professional associations.

This coincides with the ordinal successor operation for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality ( a bijection can be set up between the two by simply sending the last element of the successor to 0, 0 to 1, etc., and fixing ω and all the elements above ; in the style of Hilbert's Hotel Infinity ).

The cardinals who had elected him soon regretted their decision: the majority removed themselves from Rome to Anagni, where, even though Urban was still reigning, they elected Robert of Geneva as a rival pope on September 20 of the same year.

On the same day, the other cardinals announced that Innocent had not been canonically elected and chose Cardinal Pietro Pierleoni, a Roman whose family were the enemy of Haimeric's supporters the Frangipani.

The French king, Charles V, had recommended this, at the beginning of the schism, to the cardinals assembled at Anagni and Fondi in revolt against Urban VI, and on his deathbed he had expressed the same wish ( 1380 ).

The strict rules of the conclave were disliked by the cardinals and temporarily suspended by Pope Adrian V in 1276 before being formally revoked by John XXI's Licet felicis recordationis later that same year, both of whom had intended to promulgate new constitution governing papal election but died before doing so.

The cardinals elected this same monk as Pope Celestine V, whose main acts as pope were to reinstate the strict conclave, and to resign the papacy.

As cardinals were not allowed to vote for themselves, an elaborate procedure was adopted to ensure secrecy while at the same time prevents self voting.

In 1059, the same papal bull that restricted suffrage to the cardinals also recognised the authority of the Holy Roman Emperor, at the time Henry IV, but only as a concession made by the pope, declaring that the Holy Roman Emperor had no authority to intervene in elections except where permitted to do so by papal agreements.

New Scrutineers, Infirmarii and Revisers are not selected again after the first scrutiny ; the same nine cardinals perform the same task for the second scrutiny.

* that the same cardinals and consultors must voted in all discussions ;

However, later that same day, Enrique Almaraz y Santos, the Archbishop of Toledo, died, leaving a college of 60 cardinals to elect Pope Benedict's successor.