The Greek word aionion is translated everlasting. It is used to describe heaven and hell in Matthew 25:46. In Luke 16:22-28 there's a rich man in hell. Christ spoke of people in hell, Matthew 8:12; 22:13; 24:51; 25:30. How could a person suffer punishment if they have no conscious or they do not exist?
Is there any scriptual proof of annihilation?
One might consider;
Romans 6:23 For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord.
If eternal life is a gift from God, then it may not be with the unsaved.
One verse that brings into question the definition of eternal (aion);
Jude 1:7 Even as Sodom and Gomorrha, and the cities about them in like manner, giving themselves over to fornication, and going after strange flesh, are set forth for an example, suffering the vengeance of eternal fire.
Here it might be seen that the effects are eternal (age lasting) as opposed to the actual fire.
This is a tricky subject that often evokes strong passions.
As Beestocks said, there are different beliefs within the annihilation view.
Here are two passages that could be used to attempt to prove annihilationism:
Psalm 37:20 "But the wicked shall perish, and the enemies of the Lord shall be as the fat of lambs: they shall consume; into smoke shall they consume away." (KJV)
Matthew 10:28 "And fear not them which kill the body, but are not able to kill the soul: but rather fear him which is able to destroy both soul and body in hell." (KJV)
The Psalm says the wicked will consume away into smoke, which if you take literally, and understand to be referring to hell, can support annihilationism. And Jesus says God can "destroy" both body and soul in hell. He doesn't say "torment," although in the traditional view everyone pretty much functions as if he had.
Now, as to the question "How can it be everlasting punishment if the person is not conscious?" If you understand the punishment itself as being the cessation of existence, well, that lasts forever. If you burn up in hell and cease to exist, you cease to exist for all eternity: that's everlasting destruction. That is, you are destroyed with absolute finality. Its a punishment, and its lasts forever. It is not, however, a torment that lasts forever.
The problem here is the confusion of the terms punishment and torment, which do not have to be equivalent. Those who hold the traditional view of hell will have a hard time seeing the point that punishment doesn't have to mean torment. But annihilationists will probably be puzzled that anyone doesn't see the distinction between the two.
In terms of this world, the death penalty is a punishment, even though its torment doesn't last forever, and it is an everlasting punishment, in the sense that once you're dead you stay dead (from the worldly perspective). Annihilationism is the more absolute version of the death penalty. Because when someone is put to death by the state, their soul still exists, but annihilationism maintains that God destroys the wicked both body and soul in hell, so that they completely cease to exist. Which brings us back to Matthew 10:28, which clearly compares physical death which doesn't destroy the soul to hell which it is asserted can.
The book of revelations chapter 20 appears to debunk the idea of annihilation.
Rev 20:10 through 15 NKJV 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. And I saw a great white throne, and him that sat on it, from whose face the earth and the heaven fled away; and there was found no place for them. And I saw the dead, small and great, stand before God; and the books were opened: and another book was opened, which is the book of life: and the dead were judged out of those things which were written in the books, according to their works. And the sea gave up the dead which were in it; and death and hell delivered up the dead which were in them: and they were judged every man according to their works. And death and hell were cast into the lake of fire. This is the second death. And whosoever was not found written in the book of life was cast into the lake of fire.
Being cast into the Lake of fire is the second death the second death would of necessity have to be eternal, since death itself had already been cast into the Lake of fire.
We are also faced with the fact that first can states that that punishment is with out end. It is also important to note that death and hell gave up the those which were interred it there.
It has been my experience that those people who ascribe to total annihilation, or deny God's authority altogether, Are employing wishful thinking to Negate God's great wrath.