Does a lost soul in Hell cease to exist at some point?

Does Matthew 10:28 teach the immortality of the soul?

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.
To “destroy” a soul means a loss of everything good.  It is the loss of well-being rather than ceasing to exist.  Groups such as Christadeiphians deny the immortality of the soul.  That is, God can cause a soul to cease to exist.  I hardly think Matthew 10:28 touches on the “immortality” of the soul, but rather on its survival, which probably the Christadelphians deny too. Man can kill the body, but to kill the soul is beyond his power. “Fear Him Who can destroy (it does not say kill) both soul and body in hell.”

The word for “destroy” (apollumi) never means “to annihilate” but to render useless for its original object. Thus the same root is applied to “the lost sheep of the house of Israel” (Matthew10:6), “the marred bottles (Mark 2:22), and to the one who may “lose his life” (Luke 9:24). I believe the true doctrine of immortality has been misrepresented as well as denied. The words, “God only hath immortality” merely mean that He alone has it intrinsically, in Himself, underived, but do not deny that He communicates it to men and angels; but immortality is sometimes used as though it merely meant the faculty of never-ending existence. I believe Adam certainly had the latter at creation, but he clearly did not possess “immortality” for that means “immunity from death.” Now, had he possessed this by creation, the words, “In the day that thou eatest thereof, thou shalt surely die,” would have had no meaning, for he could not have died. It is clear he was capable of death.

The whole crux of the question is, “WHAT IS DEATH?” Is it a ceasing to exist or is it existence out of harmony with, or in separation from, God. Consider what happened when Adam sinned. He did not cease to exist. He did not die physically (that had to be provided for later by cutting him off from the tree of life). Externally and physically he was apparently unchanged. But a marked and mysterious change took place in his moral being. He knew he was naked, and he was afraid of God, and sought to hide from Him. That is, a moral gulf had yawned between him and his Creator. Death is not cessation of existence but separation of existence. This condition of death has passed on the whole human race. Man is alive to the world, but dead to God—Paul reminds the Ephesians of this their state before conversion. Then God raised them up into a new life, having quickened them together with Christ (Ephesians 2:1). This accords with what our Lord says happens to him who hears and believes. “He has eternal life . . . He has passed from death unto life” (John 5. 24). This eternal life includes immortality. The believer becomes immune from Spiritual death, but he has not yet immortality, physically speaking. That he will receive when Christ comes (see 1 Corinthians 15:).

The unbeliever will never regain immortality, but will exist in alienation and separation, physical and spiritual, from God for all eternity. “This is the second death.” That this is not a cessation of existence or annihilation of being is clear from many scriptures, e.g., “Where their worm dieth not and the fire is not quenched”; “These shall go away into everlasting punishment”; “The devil that deceived them was cast into the lake of fire and brimstone, where the beast and the false prophet are, and they shall be tormented day and night for ever and ever” (Mark 9Matthew 25:46; Revelation 20:10 and compare v 15).