Thanks for your question. You're right, Melchizedek wasn't raised during my discussions with the priest in question. So, what about Melchizedek?

First, Genesis 14:17-18, Psalm 110:4, and Hebrews 5:6, 10; 6:20; 7:1-28 give us very little information about who Melchizedek actually was. As presented in Scripture, he is rather a mystery. This is further suggested by the fact that there is no record of his beginning or ending of days (Heb. 7:3, 8). But every Pope has a beginning and ending of days. No Pope has an eternal priesthood like Melchizedek, or more specifically like Christ himself.

Second, the married Apostle Peter (Matt. 8:14; Mark 1:30; Luke 4:38; 1 Cor. 9:5) never presented himself as a priest after the order of Melchizedek. It is even disputed if Peter ever made it to Rome to be its first bishop; the Scripture doesn't record it. [1] So, how could Catholics even suggest that any Pope is after this same order of Melchizedek? Scripturally, God has made all his people a kingdom of priests before him (Exod. 19:6; 1 Pet. 2:5, 9; Rev. 1:6; 5:10), not just one mere succession of Popes, et. al.

Third, Genesis 14:18 and Hebrews 7:1 state that Melchizedek was the king of Salem, but Catholic Popes aren't kings, especially of Salem (i.e. Jerusalem). Biblically, all of God's people are kings (Dan. 7:9, 18, 22, 27; 1 Cor. 6:2-3; Rev. 20:4, 6), not just one succession of Popes; who aren't even declared by Catholics to be kings.

Fourth, there is no traceable line of the Melchizedekic Priesthood, so why would there be one in the RCC from Peter forward? Why would the traceable line begin in the N.T., not the O.T.? Jesus, who descended according to the flesh from the tribe of Judah, could not serve in the traceable Levitical Priesthood; that of Aaron (Heb. 7:14; cf. Num. 18). So, Jesus' priestly qualifications come from His belonging to the line of the Melchizedek Priesthood.

The order of Melchizedek is superior to that of the Levitical Priesthood. Christ holds it not by virtue of mere physical descent, but "by the power of an indestructible life" (Heb. 7:16; cf. Psa. 110); something no Pope can claim. All the Popes die and their bodies literally return to the dust from which they were made (Gen. 2:7; Eccl. 3:20; 12:7, etc.). While there will be a literal resurrection body for all genuine Christians at the last day (1 Cor. 15), there will also be a resurrection of the unrighteous on that same day (Dan. 12:2; Matt. 25:46; John 5:28-29; Acts 24:15).

Fifth, Abraham is recorded as only giving one "voluntary" tithe of 10% of all his plunder in war to Melchizedek (Gen. 14:20; Heb. 7:4). So, since the tithe was given as a Law to Israel and Christians are no longer under the Law (Rom. 6:14; 7:4, 6, etc.), if the Popes claim to be after the order of Melchizedek than why do they "demand" continuous monetary support from other Catholics?

The Third Council of Lateran (1179) decreed that no alienation of tithes to laymen was permissible without the consent of the Pope.

Soglia in Institut, Canon, II, 12 states, "The law of tithes can never be abrogated by prescription or custom, if the ministers of the Church have no suitable and sufficient provision from other sources; because then the natural and divine law, which can neither be abrogated not antiquate, commands that the tithe be paid."

The Catholic Code of Canon Law states, "The Christian faithful are obliged to assist with the needs of the Church so that the Church has what is necessary for divine worship for apostolic works and works of charity end for the decent sustenance of ministers" (No. 222).

[Catholic comments from NewAdvent].

By these very declarations Catholic Popes declare they are nothing like Melchizedek, or more importantly Christ himself. After-all, Popes live in the ivory tower called the Vatican with its vast wealth of treasures [2], while Christ had no where to lay his head (Luke 9:58; cf. 2 Cor. 5:1). Abraham himself was looking forward to the city that has foundations, whose designer and builder is God (Heb. 11:10; cf. Heb. 11:16; 12:22; 13:14; Rev. 21:2, 14), not some mere Popes.

So, unlike Catholic Popes, Melchizedek had no recorded family, was a priest of God Most High, blessed Abram, received one recorded voluntary tithe from Abram, was the king of Salem (king of Peace, which later became Jerusalem), his order is royal and everlasting, and he was greater than Abraham and Aaron. Popes are not descendants of the order of Melchizedek.

Footnotes:

[1] 1 Peter 5:13 states, "She who is at Babylon, who is likewise chosen, sends you greetings, and so does Mark, my son." Some maintain that "Babylon" in this text is "Rome." While this could likely be true, various identifications of the location have been suggested, among them: (1) Babylon in Egypt; (2) the ancient Mesopotamian city; or (3) Rome itself. Though Rome is called Babylon in Revelation (Rev. 17:5, 9), the interpretation that the "Babylon" in 1 Peter 5:13 is actually "Rome," remains indefinite. It is not wise to build theology upon that which is indefinite.

[2] Besides its real estate holdings, Vatican Bank's $20 million to secure its investments, and the priceless Michelangelo paintings on the ceiling of the Sistine Chapel, in 2013, the Vatican issued financial statements indicating that its holdings of gold and precious metals amounted to app. $50 million dollars.

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