Below is a brief summary of the aionios question.
Ancient Greek had two words that are common translated as “eternal”: aidios and aionios. The latter of these terms is an adjective clearly deriving from the noun aion, from which we get the English “eon”: it is an old word, appearing already in Homer, where it refers normally to a lifetime, or else some definite period of time. It never suggests an infinite stretch of time, and in later writers it continues to mean, almost always, either a lifetime or some particular period of time.
What, then, about the adjective aionios? Here is where problems arise, since the adjective seems first to occur in Plato, and Plato adapts it to a very special sense. Plato had the idea that time was a moving image of eternity, with the implication that eternity itself does not move or change: it is not an infinite length of time, but a state of timelessness (think of what time must have been like before God created the universe). This is quite different from the common meaning of aidios, which the presocratic philosophers had already used to express precisely an infinite stretch of time, with no beginning and no end; and this is what aidios continued to mean.
So, we have two adjectives in use: one of them clearly means “infinite,” when applied to time; but the other does not, and what is more, it is connected with a common noun – aion – that means simply a lifetime, with no suggestion of eternity. Aionios remains relatively rare in classical Greek, and then we come to the Septuagint, or the Greek translation of the Hebrew Bible, where it occurs very frequently (aidios, by contrast, only appears twice, and those in parts originally written in Greek). Now, aionios here can refer to things that are very old (as we say in English, “old as the hills”), but by no means eternal – what in this world is eternal? This is a very common usage, based on the Hebrew term. But it can also be used in reference to the world to come, and here we face the fundamental issue.
If one speaks of the next life, or something that happens in the next life, as aionios, does it mean simply the next era or eon, or does it carry the further implication of “eternal”? Many of the passages in the Septuagint seem to indicate that the meaning is “of that eon” – and after all, it is a very long, but still finite period of time, that elapses between our death and judgment day and the resurrection, and this could be called an era. What is more, there is some reason to think that, after the resurrection, time itself will come to an end. So, saying that punishment in the afterlife is aionios may just mean “for that eon” or epoch, and not forever.
We argued that this sense was understood by many (or most) of the Church Fathers, and that when they used aionios of punishment in the afterlife, they were not necessarily implying that punishment would be eternal. Of course, one can only show this by careful examination of specific passages in context, and this is what we tried to do in our book. Very often, the evidence is ambiguous; for example, when God is described as aionios, it is very difficult to be sure whether the word means “of the other world” or simply “eternal,” since God is both. We hope readers will decide for themselves, on the basis of the evidence we collected and the interpretations we offered.