Angelus, you make Pius IX into a heretic by claiming he taught what he did not actually teach. What you claim he teaches contradicts Church Dogma as it has "once been declared" (again, ~8 times). So, please, stop.
Read what the Church has infallibly taught regarding salvation first, and then apply it to what Pius IX taught. You will not misintepret him that way. You are going in the opposite order, taking your misinterpretation of Pius IX and applying it to Church Dogma. That is what leads to what you originally posted in this thread, which was a word-for-word denial of the Dogma. Please, read what has infallibly been taught regarding EENS. We must keep the faith "whole and undefiled" or we will, without doubt, perish eternally

As I've tried other explain to you, Pius IX and I have not misinterpreted the dogma. 

As I've tried other explain to you, Pius IX and I have not misinterpreted the dogma. You have. Those who are "invincibly ignorant of our most holy religion" AND who follow "the natural law... illumined by divine light and grace" in their moral actions, will not "suffer eternal punishments." Those are the exact words of Pius IX.

That does not mean that there will be many people "attain eternal life" in the way the Pius IX describes. I don't know for sure, of course. But I would guess there are very few people. Think of Abel and Lot and Noah. Very, very few men live their lives in the way required by the "invincible ignorance" teaching.

2 Corinthians 4:3 And if our gospel be also hid, it is hid to them that are lost

Pope Pius IX, First Vatican Council, Sess. 3, Chap. 2 on Revelation, 1870, ex cathedra: “Hence, also, that understanding of its sacred dogmas must be perpetually retained, which Holy Mother Church has once declared; and there must never be a recession from that meaning under the specious name of a deeper understanding.

For ‘there is one universal Church outside of which no one at all is saved; it contains regular and secular prelates along with those under their jurisdiction, who all profess one Lord, one faith and one baptism.”

Pope Pius IX, Ubi primum

For, in truth, when released from these corporeal chains, ‘we shall see God as He is’ (1 John 3:2), we shall understand perfectly by how close and beautiful a bond divine mercy and justice are united; but, as long as we are on earth, weighed down by this mortal mass which blunts the soul, let us hold most firmly that, in accordance with Catholic teaching, there is ‘one God, one faith, one baptism’ [Eph. 4:5]; it is unlawful to proceed further in inquiry

Pope Pius IX, Singulari Quadam

Pope Innocent III, Fourth Lateran Council, Constitution 1, 1215, ex cathedra: “There is indeed one universal Church of the faithful, outside of which nobody at all is saved, in which Jesus Christ is both priest and sacrifice.”

Pope Boniface VIII, Unam Sanctam, Nov. 18, 1302, ex cathedra:
“With Faith urging us we are forced to believe and to hold the one, holy, Catholic Church and that, apostolic, and we firmly believe and simply confess this Church outside of which there is no salvation nor remission of sin… Furthermore, we declare, say, define, and proclaim to every human creature that they by absolute necessity for salvation are entirely subject to the Roman Pontiff.”

Pope Clement V, Council of Vienne, Decree # 30, 1311-1312, ex cathedra:
“Since however there is for both regulars and seculars, for superiors and subjects, for exempt and non-exempt, one universal Church, outside of which there is no salvation, for all of whom there is one Lord, one faith, and one baptism…”

Pope Eugene IV, Council of Florence, Sess. 8, Nov. 22, 1439, ex cathedra:
“Whoever wishes to be saved, needs above all to hold the Catholic faith; unless each one preserves this whole and inviolate, he will without a doubt perish in eternity.”

Pope Eugene IV, Council of Florence, “Cantate Domino,” 1441, ex cathedra:
“The Holy Roman Church firmly believes, professes and preaches that all those who are outside the Catholic Church, not only pagans but also Jews or heretics and schismatics, cannot share in eternal life and will go into the everlasting fire which was prepared for the devil and his angels, unless they are joined to the Church before the end of their lives; that the unity of this ecclesiastical body is of such importance that only for those who abide in it do the Church’s sacraments contribute to salvation and do fasts, almsgiving and other works of piety and practices of the Christian militia produce eternal rewards; and that nobody can be saved, no matter how much he has given away in alms and even if he has shed blood in the name of Christ, unless he has persevered in the bosom and unity of the Catholic Church.”

Pope Leo X, Fifth Lateran Council, Session 11, Dec. 19, 1516, ex cathedra:
“For, regulars and seculars, prelates and subjects, exempt and non-exempt, belong to the one universal Church, outside of which no one at all is saved, and they all have one Lord and one faith.”

Pope Pius IV, Council of Trent, “Iniunctum nobis,” Nov. 13, 1565, ex cathedra: “This true Catholic faith, outside of which no one can be saved… I now profess and truly hold…

Pope Pius IX, Vatican Council I, Session 2, Profession of Faith, 1870, ex cathedra: “This true Catholic faith, outside of which none can be saved, which I now freely profess and truly hold…

Pope St. Gregory the Great, quoted in Summo Iugiter Studio, 590-604:
“The holy universal Church teaches that it is not possible to worship God truly except in her and asserts that all who are outside of her will not be saved.”

Pope Innocent III, Eius exemplo, Dec. 18, 1208:
“By the heart we believe and by the mouth we confess the one Church, not of heretics, but the Holy Roman, Catholic, and Apostolic Church outside of which we believe that no one is saved.”

Pope Clement VI, Super quibusdam, Sept. 20, 1351:
“In the second place, we ask whether you and the Armenians obedient to you believe that no man of the wayfarers outside the faith of this Church, and outside the obedience to the Pope of Rome, can finally be saved.”

Pope St. Pius V, Bull excommunicating the heretic Queen Elizabeth of England, Feb. 25, 1570: “The sovereign jurisdiction of the one holy Catholic and Apostolic Church, outside of which there is no salvation, has been given by Him [Jesus Christ], unto Whom all power in Heaven and on Earth is given, the King who reigns on high, but to one person on the face of the Earth, to Peter, prince of the Apostles... If any shall contravene this Our decree, we bind them with the same bond of anathema.

Pope Leo XII, Ubi Primum (# 14), May 5, 1824:
“It is impossible for the most true God, who is Truth itself, the best, the wisest Provider, and the Rewarder of good men, to approve all sects who profess false teachings which are often inconsistent with one another and contradictory, and to confer eternal rewards on their members… by divine faith we hold one Lord, one faith, one baptism… This is why we profess that there is no salvation outside the Church.

Pope Leo XII, Quod hoc ineunte (# 8), May 24, 1824: “We address all of you who are still removed from the true Church and the road to salvation. In this universal rejoicing, one thing is lacking: that having been called by the inspiration of the Heavenly Spirit and having broken every decisive snare, you might sincerely agree with the mother Church, outside of whose teachings there is no salvation.

Pope Gregory XVI, Mirari Vos (# 13), Aug. 15, 1832: “With the admonition of the apostle, that ‘there is one God, one faith, one baptism’ (Eph. 4:5), may those fear who contrive the notion that the safe harbor of salvation is open to persons of any religion whatever. They should consider the testimony of Christ Himself that ‘those who are not with Christ are against Him,’ (Lk. 11:23) and that they disperse unhappily who do not gather with Him. Therefore, ‘without a doubt, they will perish forever, unless they hold the Catholic faith whole and inviolate (Athanasian Creed).”

Pope Gregory XVI, Summo Iugiter Studio (# 2), May 27, 1832:
“Finally some of these misguided people attempt to persuade themselves and others that men are not saved only in the Catholic religion, but that even heretics may attain eternal life.”

Pope Pius IX, Nostis et Nobiscuм (# 10), Dec. 8, 1849: “In particular, ensure that the faithful are deeply and thoroughly convinced of the truth of the doctrine that the Catholic faith is necessary for attaining salvation. (This doctrine, received from Christ and emphasized by the Fathers and Councils, is also contained in the formulae of the profession of faith used by Latin, Greek and Oriental Catholics

Pope Leo XIII, Tametsi futura prospicientibus (# 7), Nov. 1, 1900: “Christ is man’s ‘Way’; the Church also is his ‘Way’… Hence all who would find salvation apart from the Church, are led astray and strive in vain

Pope St. Pius X, Iucunda sane (# 9), March 12, 1904: “Yet at the same time We cannot but remind all, great and small, as Pope St. Gregory did, of the absolute necessity of having recourse to this Church in order to have eternal salvation…”

Pope St. Pius X, Editae saepe (# 29), May 26, 1910: “The Church alone possesses together with her magisterium the power of governing and sanctifying human society. Through her ministers and servants (each in his own station and office), she confers on mankind suitable and necessary means of salvation.”

Pope Pius XI, Mortalium Animos (# 11), Jan. 6, 1928: “The Catholic Church is alone in keeping the true worship. This is the fount of truth, this is the house of faith, this is the temple of God: if any man enter not here, or if any man go forth from it, he is a stranger to the hope of life and salvation.

Abraham lived prior to the revelation of the divine law, given to Moses. He and people of the Age of the Patriarchs (aka Fathers) lived according to the natural law illumined by the divine grace. The pagans who are/were "invincibly ignorant of our holy religion," the Catholic faith, AND also live/lived according to the natural law illumined the divine grace.

As Pius IX said, "Because God knows, searches and clearly understands the minds, hearts, thoughts, and nature of all, his supreme kindness and clemency does not permit anyone at all who is not guilty of deliberate sin to suffer eternal punishments."

These are questions that it might be worth discussing among those of good will here to believe the Church's dogmatic teaching on EENS, and we should stop wasting our time on trolling heretics like Angelus.

Just as St. Thomas Aquinas fully articulated the distinction between natural punishment / natural happiness, rooted in justice, and supernatural beatitude in the Kingdom, something above human nature, which, being owed to no one and not required for happiness, there would be no injustice in not granting it, since it's a completely free gift ...

similarly, I believe that there's confusion between a natural justification and a supernatural one, where infants in Limbo, for instance, are not in a "state of grace", meaning that they do NOT possess the supernatural virtues of faith, hope, and charity (where of course only charity remains in eternity, as the Holy Ghost teaches through St. Paul).

Since these innocent souls have committed no actual sin, they are not in a state of enmity with God, which is usually what's contrasted with justification, except that what people often mean by justification is ... being in a state of grace, and they're not in that state either.  So they're neither justified nor non-justified.  They're in a sort of natural friendship with God, where they believe in Him, obey Him, assent to His truths and to the law He has written in their hearts (the natural law) ... in a natural way only, since they do not and cannot have supernatural faith and supernatural charity.

So when Pax says there can be no justification except in the Church, he's assuming the definition where justification means being in a state of grace and having the supernatural virutes, in which sense he's correct.  BUT ... that does not mean there can't be a certain NATURAL type of justificaiton, a natural friendship with God.  That is where the post-Tridentine theologian Melchior Cano, OP stated that infidels can be justified but not saved, where he's referring to a type of justification that can be achieved short of having the supernatural virtues, since infidels, lacking faith, cannot have the supernatural virtues.  Theologians who discussed Trent on Justification stated that there are natural virtues they they call "fides initialis", "caritas initalis", "spes initialis", which are in fact natural analoguest to the respective supernatural virtues, and they can in fact develop these natural virtues, the natural equivalents of the supernatural ones by the same name, nay, they MUST develop these natural virtues in order to be led to the very point of receiving the SUPERNATURAL ones, which then only happens at the actual reception of the Sacrament of Baptism, which confers the supernatural virutes ex opere operato.  So while these natural virute are necessary dispositions to receive the freely-given unmerited supernatural equivalents, they do not somehow natural lead to, transition into or somehow merit the gift of the supernatural ones.

By Divine Law, you mean Moses and the 10 commandments?  You think the Israelites and others before Moses didn’t know about the moral law? 

You're denying God's revelation to Adam/Eve of a coming Redeemer, which ALL THE FATHERS knew of, prior to Moses.  Adam, Noe, Abraham, Isaac, Jacob, etc. 

These were not pagans or comparable to pagans.  You should be ashamed. 

These are questions that it might be worth discussing among those of good will here to believe the Church's dogmatic teaching on EENS, and we should stop wasting our time on trolling heretics like Angelus.

Just as St. Thomas Aquinas fully articulated the distinction between natural punishment / natural happiness, rooted in justice, and supernatural beatitude in the Kingdom, something above human nature, which, being owed to no one and not required for happiness, there would be no injustice in not granting it, since it's a completely free gift ...

similarly, I believe that there's confusion between a natural justification and a supernatural one, where infants in Limbo, for instance, are not in a "state of grace", meaning that they do NOT possess the supernatural virtues of faith, hope, and charity (where of course only charity remains in eternity, as the Holy Ghost teaches through St. Paul).

Since these innocent souls have committed no actual sin, they are not in a state of enmity with God, which is usually what's contrasted with justification, except that what people often mean by justification is ... being in a state of grace, and they're not in that state either.  So they're neither justified nor non-justified.  They're in a sort of natural friendship with God, where they believe in Him, obey Him, assent to His truths and to the law He has written in their hearts (the natural law) ... in a natural way only, since they do not and cannot have supernatural faith and supernatural charity.

So when Pax says there can be no justification except in the Church, he's assuming the definition where justification means being in a state of grace and having the supernatural virutes, in which sense he's correct.  BUT ... that does not mean there can't be a certain NATURAL type of justificaiton, a natural friendship with God.  That is where the post-Tridentine theologian Melchior Cano, OP stated that infidels can be justified but not saved, where he's referring to a type of justification that can be achieved short of having the supernatural virtues, since infidels, lacking faith, cannot have the supernatural virtues.  Theologians who discussed Trent on Justification stated that there are natural virtues they they call "fides initialis", "caritas initalis", "spes initialis", which are in fact natural analoguest to the respective supernatural virtues, and they can in fact develop these natural virtues, the natural equivalents of the supernatural ones by the same name, nay, they MUST develop these natural virtues in order to be led to the very point of receiving the SUPERNATURAL ones, which then only happens at the actual reception of the Sacrament of Baptism, which confers the supernatural virutes ex opere operato.  So while these natural virute are necessary dispositions to receive the freely-given unmerited supernatural equivalents, they do not somehow natural lead to, transition into or somehow merit the gift of the supernatural ones.

You beg the question that anyone at all who follows the natural law dies invincibly ignorant. That is not what Pius IX said. Aquinas proposed that they will be instructed in the faith prior to death

At the time of the apostles it was taught that the Gospel had  been preached to the ends of the Earth:

You deny the Dogma:

Pope Eugene IV: "all those who are outside the Church...cannot share in eternal life and will go into the everlasting fire which was prepared for the devil and his angels"

You: "not all who are outside the Church will burn in hell forever and never make it to paradise"

You say those who die outside the Church can "make it to paradise". "Paradise" is the Kingdom of Heaven, the Beatific Vision. None of those outside of the Church can partake in it

Read the Athanasian creed as "once declared", believe it, and then profess it.

You must convert to the Catholic Faith if you wish to be saved