I'd like to address this one, Benzoic. It is perhaps the most enduring argument for "eternal punishment". Indeed, Augustine, in the late fourth century, used it ----- Augustine, who doubtless influenced the later church more than any other. Augustine had been involved with a cult which believed that there would be an eternal struggle between good and evil. It seems that Augustine brought a form of that concept into the church by promoting spreading his teaching about eternal torment."And these will go away into eternal punishment, but the righteous into eternal life" (Mat. 25:46) I’m aware that the Greek word translated as eternal, to some, does not necessarily mean eternal. However, why does the word ‘eternal’ life mean exactly that, eternal life, while ‘eternal’ punishment does not mean eternal punishment? The same Greek word used for ‘eternal’ punishment is used for ‘eternal’ life. Why then would you assume that the punishment in hell is not eternal?
You are quite right in saying that if "aiōnios" means "eternal" in the case of the sheep, it will also mean "eternal" in the case of the goats. However, there's another little conundrum involved in the statement of Christ as recorded by Matthew. The Greek word which has been translated as "punishment" actually means "correction". In its verbal form, the word means "prune". When a tree is pruned, its growth is corrected.
Now how can one have "eternal" correction? If correction is accomplished, then it is not eternal.
Actually "aiōnios" never means "eternal". It is the adjectival form of the noun "aiōn" which means "age", and from which is derived our English word "æon"(age).Indeed, there is even an English adjective corresponding to "aiōnios", namely "æonian" whose meaning in dictionaries is given as "pertaining to an æon".
"Aiōnios" seems to mean "going from age to age" or sometimes "permanent". Of course we all know that "permanent" does not usually mean "forever".
So the goats will be given a correction which goes from age to age, and the sheep enter into life which goes from age to age. Now that does not imply that their age-to-age life will come to an end. "Aiōnios" does not itself contain the idea of "eternal". So that which is "aiōnios" may come to an end or may not.