New International Version, Matthew 25:
Do these two uses of "eternal" carry the same meaning? If yes, what does it mean?
They are the same Greek word in the same sentence, so I think they carry the same meaning.
Eternal life is immortal life.
The question is this: What is eternal punishment?
The term
eternal punishment is the opposite of eternal life. The punishment is eternal
death, i.e., annihilation or the absence of life, Revelation 20:
Conscious punishment or suffering over an infinite period of existence makes little physical sense. To be conscious, you need a life or a breadth from God (Genesis 2:7). In other words, conscious torment over an infinite time is a form of eternal 'life'.
My paraphrase:
Paul expressed a similar concept in 2 Thessalonians 1:
Strong's Greek: 3639. ὄλεθρος (olethros) — 4 Occurrences
BDAG:
① a state of destruction, destruction, ruin, death in our lit. always w. some kind of transcendent coloring, … ὄλ. αἰώνιος
eternal death (TestReub 6:3) 2 Th 1:9
Pulpit Commentary:
There is some justification that eternal punishment is permanent death or annihilation of the soul.
See also
Does the Bible teach conscious torment over an infinite period?.