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Of course, in the Greek, Mat. 16 doesn't say what you want it to say. The Lord makes a distinction between Petros (a small stone, a piece of a stone) and Petra (a large rock). Besides which, the Scriptures are clear as to who the Rock on whom we are built is:

[God] is the Rock; His work is perfect. For all His ways are just, a God of faithfulness, and without evil; just and upright is He.
(Deuteronomy 32:4)
But Jeshurun grew fat and kicked. You grew fat, thick, and satisfied. Then he forsook God who made him, and lightly esteemed the Rock of his salvation.
(Deuteronomy 32:15)
YHVH is my strength, and my fortress, and my deliverer; my God, my rock; I will trust in Him; He is my shield, and the horn of my salvation, my high tower.
(Psalms 18:2)
For who is God besides YHVH? Or, who is a Rock except our God?
(Psalms 18:31)
Because you have forgotten the God of your salvation, and have not been mindful of the Rock of your strength, therefore you shall plant pleasant plants and shall sow it a fresh shoot.
(Isaiah 17:10)
. . . therefore so says the Lord YHVH, "Behold, I place in Zion a Stone for a foundation, a tried Stone, a precious Cornerstone, a sure Foundation; he who believes shall not be ashamed."
(Isaiah 28:16)
Therefore whoever hears these sayings of Mine, and does them, I will liken him to a wise man who built his house on a rock.
(Matthew 7:24)
For any other foundation can no one lay than that which is laid, which is Yeshua HaMashiach.
(I Corinthians 3:11)
And all were baptized to Moses in the cloud and in the sea, and all ate the same spiritual food, and all drank the same spiritual drink; for they drank of the spiritual Rock that followed them, and that Rock was the Messiah.
(I Corinthians 10:2-4)
For having been drawn to Him, a living Stone, indeed rejected by men, but elect, precious with God; you also as living stones are built up a spiritual house, a holy priesthood, to offer up spiritual sacrifices acceptable to God through Yeshua HaMashiach. Therefore also it is contained in the Scripture: "Behold, I lay in Zion a chief corner Stone, elect, precious, and he who believes on Him shall never be ashamed." Therefore to you who believe is the honor. But to those who are disobedient, He is the Stone which the builders rejected; this One came to be the Head of the corner, and a Stone-of-stumbling and a Rock-of-offense to those disobeying, who stumble at the word, to which they also were appointed.
(I Peter 2:4-8)

When the Bible calls God our Rock literally dozens of times (the above is but a sample) and calls Kefa (Peter) a rock but once, the correct understanding is clear: The true Church is built not upon Kefa or his supposed successors (which would have to be chosen by laying on of hands, not election by the cardinals in any case, as another poster pointed out), but upon Yeshua the Messiah, God Incarnate, the sole Rock of our Salvation.