I'll offer a Latter-day Saint's perspective
Eternal man
We have always existed and will always exist (D&C 93:29,33; Alma 42:16).
We have a "self" (or "mind" or "intelligence" or "consciousness") that has been from eternity past; it was clothed with a spirit by God our Father. (Abraham 3:22-23, Moses 3:7, Acts 17:29, Family Proclamation par. 3)
Upon entering mortal life, we were further clothed with a physical body: spirit and body were joined together. Spirit (and "intelligence") continue on in the Spirit World after physical death; the body molders on earth. (Genesis 2:7, Eccl. 12:7, Alma 40:11-14)
At the resurrection, spirit & body are reunited, never again to be separated. All members of the family of Adam will exist forever with an immortal spirit joined to an immortal body. (D&C 138:17, 2 Nephi 9:21-22)
We did not begin to exist; we will not cease to exist--rather, our "self" is transformed through the process known as the Plan of Salvation. I see a beginning & ending of our existence as category errors--that's not the kind of beings we are are--we are eternal beings, clothed with spirit & body in the image of God.
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What are life & death
Life is a joining together; death is a separation. Although new life may change our state of being/experience/perceptions, it does not begin our consciousness. Death may change our state of being/experience/perceptions, but it does not end our consciousness.
The (Nephite) prophet Jacob explained death:
10 O how great the goodness of our God, who prepareth a way for our
escape from the grasp of this awful monster; yea, that monster, death
and hell, which I call the death of the body, and also the death of
the spirit.
11 And because of the way of deliverance of our God, the Holy One of
Israel, this death, of which I have spoken, which is the temporal,
shall deliver up its dead; which death is the grave.
12 And this death of which I have spoken, which is the spiritual
death, shall deliver up its dead; which spiritual death is hell;
wherefore, death and hell must deliver up their dead, and hell must
deliver up its captive spirits, and the grave must deliver up its
captive bodies, and the bodies and the spirits of men will be restored
one to the other; and it is by the power of the resurrection of the
Holy One of Israel. (2 Nephi 9:10-12)
He describes physical death in verse 7:
this flesh must have laid down to rot and to crumble to its mother
earth
He describes spiritual death in verse 9:
to be shut out from the presence of our God, and to remain with the
father of lies, in misery, like unto himself
To borrow a phrase from SpiritRealmInvestigator: "death" & "consciousness" are orthogonal properties.
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Kingdoms of Glory
I do not believe salvation/damnation are boolean concepts. Paul spoke of different degrees of glory among resurrected bodies (see 1 Cor. 15:20-22); through Joseph Smith additional details were revealed:
(source)
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Outer Darkness
32 They are they who are the sons of perdition, of whom I say that it
had been better for them never to have been born;
33 For they are vessels of wrath, doomed to suffer the wrath of God,
with the devil and his angels in eternity;
34 Concerning whom I have said there is no forgiveness in this world
nor in the world to come—
35 Having denied the Holy Spirit after having received it, and having
denied the Only Begotten Son of the Father, having crucified him unto
themselves and put him to an open shame.
36 These are they who shall go away into the lake of fire and
brimstone, with the devil and his angels—
37 And the only ones on whom the second death shall have any power (D&C 76:32-37)
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Telestial
(often associated with the glory of the present world)
81 And again, we saw the glory of the telestial, which glory is that
of the lesser, even as the glory of the stars differs from that of the
glory of the moon in the firmament.
82 These are they who received not the gospel of Christ, neither the
testimony of Jesus.
83 These are they who deny not the Holy Spirit.
84 These are they who are thrust down to hell.
85 These are they who shall not be redeemed from the devil until the
last resurrection, until the Lord, even Christ the Lamb, shall have
finished his work. (D&C 76:81-85)
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Terrestrial
(often associated with the glory of the earth in its Edenic state)
75 These are they who are honorable men of the earth, who were blinded
by the craftiness of men.
76 These are they who receive of his glory, but not of his fulness.
...
78 Wherefore, they are bodies terrestrial, and not bodies celestial,
and differ in glory as the moon differs from the sun.
79 These are they who are not valiant in the testimony of Jesus;
wherefore, they obtain not the crown over the kingdom of our God.
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Celestial
(the glory of God)
51 They are they who received the testimony of Jesus, and believed on
his name and were baptized after the manner of his burial, being
buried in the water in his name, and this according to the commandment
which he has given—
52 That by keeping the commandments they might be washed and cleansed
from all their sins, and receive the Holy Spirit by the laying on of
the hands of him who is ordained and sealed unto this power;
53 And who overcome by faith, and are sealed by the Holy Spirit of
promise, which the Father sheds forth upon all those who are just and
true.
...
62 These shall dwell in the presence of God and his Christ forever and
ever.
...
69 These are they who are just men made perfect through Jesus the
mediator of the new covenant, who wrought out this perfect atonement
through the shedding of his own blood.
70 These are they whose bodies are celestial, whose glory is that of
the sun, even the glory of God, the highest of all...
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Eternal
Eternal is a descriptor of God. I have written more extensively on the topic here.
10 For, behold, the mystery of godliness, how great is it! For,
behold, I am endless, and the punishment which is given from my hand
is endless punishment, for Endless is my name. Wherefore—
11 Eternal punishment is God’s punishment.
12 Endless punishment is God’s punishment. (D&C 19:10-12)
Jesus paid the price of our sins to satisfy the demands of justice and extend mercy:
15 And now, the plan of mercy could not be brought about except an
atonement should be made; therefore God himself atoneth for the sins
of the world, to bring about the plan of mercy, to appease the demands
of justice, that God might be a perfect, just God, and a merciful God
also.
16 Now, repentance could not come unto men except there were a
punishment, which also was eternal as the life of the soul should be,
affixed opposite to the plan of happiness, which was as eternal also
as the life of the soul. (Alma 42:15-16)
If we reject the mercy offered by Jesus, there is an eternal punishment:
15 Therefore I command you to repent—repent, lest I smite you by the
rod of my mouth, and by my wrath, and by my anger, and your sufferings
be sore—how sore you know not, how exquisite you know not, yea, how
hard to bear you know not.
16 For behold, I, God, have suffered these things for all, that they
might not suffer if they would repent; (D&C 19:15-16)
Eternal here is describing God specifically, not duration generally:
- Eternal punishment is God's punishment
- Eternal life is God's life
Immortality describes living forever. To quote Dr. Jason Carroll:
Eternal life is a life that is both endless in duration and godlike in
quality
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Conclusion
God's plan of salvation will enable His children to become the very most they are willing to become. In this sense salvation, in the fullest and most complete sense of the word, describes the result of fully participating in the plan of salvation.
A deeper dive on my thoughts on the meaning of salvation is found here.
Those separated from the source of eternal life (God) will not have eternal life--their eternal progression will be halted, there's an asymptote in their future.
How can the wicked live for eternity in hell when they are completely separated from the only source of eternal life?
My conclusion follows if we allow that death & consciousness are orthogonal properties (otherwise this statement is obviously not an answer to the question):
I have not argued that the wicked will live forever in hell, but that they will live forever. They will not, however, have eternal life, the kind of life that God has.
Disclaimer: these thoughts are the product of my own study and do not constitute official statements by The Church of Jesus Christ of Latter-day Saints
"D&C" in the citations above refers to the book of Doctrine & Covenants