Matthew 25:46 KJV: And these shall go away into everlasting punishment; but the righteous into life eternal.
In the Greek;
Kai apeleusontai houtoi eis kolasin aionion hoi de diaioi eis zoen aionion.
concordant Literal Translation;
And these shall be coming away into chastening eonian, yet the just into life eonian.
This passage is the most commonly seen verse when dealing with Aionios, or Aionion. Aionios is a word that is difficult to translate, or render into English especially since our understanding of it in the idea of “eternal” or “everlasting” has become equated with ceaselessness or endlessness in duration, at least on the part of punishment and life. Now certainly the life is immortal, but what about the punishment? Of course there is always the “kolasin” argument, that being that it means chastisement. But what I want to do here is not to talk about the chastisement or life per’se, but to offer a translation of Aionios/n, as best as my study and understanding of it can render itself. I want to offer it in a way that it might make readable sense and be more clear, if I can. I have presented this idea elsewhere, but here I’d like to present it also.
I would also like some help in finding a more aesthetic way to render it once the literal rendering is given. 
So here it goes, my rendering and the logic behind it;
And these shall depart into chastisement [chastisement lasting] yet the righteous into life [life lasting].
The logic behind this, is understanding that aionion/aionios is an adjective. An adjective describes a noun, but it does not have power that extends beyond the already present ‘nature-of-fact’ (as I might try to term it) in that noun.
Take for example this sentence;
“The Titanic is long” - long here being the adjective. Now the Titanic is a long boat of course, but how long is it? Is it as long as the continent of Africa? Truthfully to say that would be to either lie or to engage in metaphor. The Titanic cannot be by ‘nature-of-fact’ or ‘in itself’ as long as the continent of Africa. However, it is still a long or large ship. The sentence “The Titanic is long” is a distilled form of what would literally translate out as;
“The Titanic (the ship named the Titanic) is the length of the Titanic, of itself, the ship named the Titanic; the Titanic is as long as itself.”
Or
“The Titanic is Titanic long.” This is a true sentence, and faithfully keeps the nature of the adjective “long” without any detriment to the truth and neither does it bring with it any misunderstanding of the length (that I know of). So it is with Aionios in the way I rendered the verse, and here are some of the truths to be taken out of the way I rendered the verse, I feel, that do not in anyway degrade the higher beliefs and hopes of the faithful.
Chastisement [chastisement lasting] The chastisement lasts, until it lasts. It does not fling the timeline or endurance of the chastisement beyond the ‘nature-of-fact’ in the noun, but it does not set a limiter on it either below the noun’s power to achieve justice. In other words; “Chastisement [chastisement lasting]” is a chastisement that will last until it has achieved its end, and achieved the justice and correction the noun implies.
Life [life lasting] The life lasts as long as life lasts. Now, understanding that life here is God’s life, that is; Life lasting, or Jesus lasting. Jesus is the Life after all. So life that is Life lasting is a life that is by default a life of immortality, endless as God is endless, and full of God’s quality. There is no degredation or “making temporary” the Life that we are promised, for the temporal nature of the chastisement mentioned before, and the translation remains faithful to the ‘nature-of-fact’ in the noun, as well as strengthens it making the life a self-sufficient life, and self-describing life grammatically.
This therefore is the translation I offer, and the method of understanding how it came about. Any thoughts and suggestions, especially aesthetic ways to render it (if it is not pleasing or pretty enough as it is) are more than welcome. Discussion is also welcome.
God bless, and I hope this helps with understanding Aionios/n.
I think that the fact that Dan 7:14 explains aionios (in Septuagint Greek; in the original, it’s the Aramaic form of “olam,” meaning “remote time”; considered to be the Hebrew version of aionios) shows that the word often needs clarification. In this instance Daniel understood what it meant and wanted to leave no room for misunderstanding. Therefore, he explains that what he means by “everlasting” (“alam” or “olam”) is not just “super-impressive, divine, and unclear as to when it will end,” but “in fact will never be subject to fading away, nor will it ever be capable of being destroyed.” Which is exactly what we would hope for and expect from the reign of the Son of Man!