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The Mormons use this verse as “evidence” to support their belief in a Celestial Kingdom and a Terrestrial Kingdom in Heaven. How does the Catholic Church explain this verse?

The Mormons are partially correct in stating that there are multiple kingdoms. For even the Catholic church holds that there is heaven and hell (peona damni…sp?).
The one is the kingdom of God, and the other is the kingdom of the damned.
St. Paul, here, is concerned with foolish questions – so, understand, he is contrasting a foolish notion that bodies are NOT raised with a corrected one. The factions in the Jewish community (Sadducee and Pharisee) held opposing doctrines concerning the resurrection – that is, the Sadducee held that the dead are not raised.
see: Matthew 22:23-33
Clearly, some at Corinth had been deceived by the Sadducee apologetic or one very similar to it.
Notice that the topic of St. Paul is that bodies are resurrected and so he must be giving a REASON that the foolish ideas are in fact foolish.
In the last verses of Matthew, Jesus notes to the
crowd’s astonishment that God said “I
am the God of Abraham, Isaac, and Jacob.”
God’s statement means that, even without a body, the fathers are yet alive.
Because, God says this of the fathers
after their earthly bodies are
in the grave – eg: that he posses them.
eg:I
am the God.
God also says this so very distinctly by NOT referring to “Israel” which could be taken to be the entire nation – rather he uses the curious name “Jacob” – eg: Is-ra-el’s name BEFORE he did evil (Got the better of) God.
This saying is quite mysterious, and is in the indisputable book of Moses which even the anathema Sadducee held as binding. It refers to one of several possibilities – but setting later Thomistic ideas aside for a moment, the proto materialist Sadducee and their pagan equivalent (the Epicuran ) were grappling with the question of “where did God come from” – is “God” made of matter? (pantheistic).
Rather than bring all the details forward, let’s take a look at 1Corinthians again, and recall that St. Paul speaks of the Analogy of the body in other places – that is, we
are the body of Christ, and Jesus himself who says "I
am the resurrection.
Perhaps you begin to see the consistency?
They are physically dead “in” god to the pantheist,
BUT living according to Jesus and the revelation of God to Moses. That God would take on flesh, himself, (to raise it) was simply beyond many of the materialistic listeners. (

… I was listening to my sons science lecture – which stated, “everything is made of matter” – So, It still exists, today…)
This is the difference in necromancy, the calling upon the damned, and prayer to saints alive in the body of Jesus.
Now, look again at 1Corinthians, and the consistency ought to shine forth –
In V:15:32, St. Paul speaks of the Ephesian “beasts” or “animals” – far more than just name calling, St. Paul shows forth his judgment of those who do not believe in Jesus.
Hardly a coincidence, then, that Paul contrasts the flesh of animals with that of man, eg: the necessarily corruptible with that which may be resurrected in verse 15:39.
I think the Douay commentary falls short here, in that it focuses on the Glory of the resurrected body only – and not the eternal shame of those raised to judgment. Those who’s bodies will remain like those of the animals – the dust which the devil may feast on. Satan’s body.
There is also an allusion, which I do not fully understand, to the taxonomy of flesh into four specific kinds. Birds and fish being beyond my understanding at the moment.
But, that St. Paul is classifying bodies is what is important to grasp – not “flesh” when speaking of the resurrection of Jesus.
Note back in 1Corinthians 15:22 that “in Christ shall all be made alive” – His body.
And echoing back to the passage of Moses where the bodies had already died, the mystery of their life is tied again to the body of God. (not pantheistic, though).
That this union is what St. Paul has in mind, that of a man and wife who BECOME one flesh is attested one more time in 1Cori15:25-28 – a passage about subjection to the Lord, as Eve (life) was subject to Adam(Lord).
Eve was told she would return to her man, and also that she would be saved through her children. St. Paul is trying to say the same thing, over and over, in hopes of getting the message across in at least one way.
Thus, there is dirt under Christs feet – whether it be hell under the Jerusalem temple, or (in analogy) from his throne in heaven with his feet standing on the dust of the earth.
St. Paul, I think sums the theology of the body up the most when he says: 1Cori 15:28 – that God will be the
all in every thing. The marriage of God with his creation and the obedience of creation to God out of love or, as necessary, out of conquest.
KJV
Revel 20:10 And the devil that deceived them was cast into the lake of fire and brimstone, where the beast and the false prophet are, and shall be tormented day and night for ever and ever.
Modified KJV:
Revel 21:3 And I heard a great voice out of heaven saying, Behold, the
tent of God is with men, and he will dwell with them, and they shall be his people, and God himself shall be with them, and be their God.
I don’t know what apologetic you have heard from the Mormons, but the images of St. Paul are consistent with his analogy of the body and marriage, and one family. Father, Son, and bride. Perhaps you have specific questions no one has yet answered?