Augustine's reasoning concerning the meaning of aionian in Matthew 25:46 (though followed by most theologians since) is clearly fallacious, because it is simply not true that unless aionian signifies endless duration there (in the context of life and chastening) we're left with no scriptural promise of unending life for the saints. Augustine (and those who follow him) ignore the fact that whatever aionian life and aionian punishment mean in Matthew 25:46, we still have a clear promise of unending life in passages like Luke 20:34-36. Where Christ said "...The children of this world (Greek: age, eion) marry, and are given in marriage: But they that shall be accounted worthy of that world (Greek age, eion), and of the resurrection from the dead, shall neither be married, nor take wives. Neither can they die any more: for they are equal to the angels, and are the children of God, being the children of the resurrection." And we're still told that God is immortal in passages like Romans 1:26. So Augustine's fears of what would be lost if we don’t understand aionian to mean unending were (and are) unfounded. And his view is even more fallacious when it's realized that unending duration was not what the Greek speaking world understood aionian, aionios, and aionas to mean in the Pratristic era. The following quote from a non-Christian (neo-Platonist) writing in the 6th century shows this.
Do not suppose that the soul is punished for endless eons (apeirou aionas) in Tartarus. Very properly, the soul is not punished to gratify the revenge of the divinity, but for the sake of healing. But we say that the soul is punished for an aionion period (aionios) calling its life and its allotted period of punishment, its aeon.
Source: Olnmpiodorus the Younger (AD 550)
And this quote from an 8th century commentary on Holy Scripture, shows that centuries later, even the Latin words used to translate aionios, aionian, and aionas in the Latin Vulgate were not necessarily understood to mean endless durion.
Ambrosius Autpertus (An 8th century Bennedictine monk, abbot, and Theologian who wrote commentaries on Holy Scripture.)
Rev 20:10 “And 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 (Latin saecula saeculorum)." But when it says, "they shall be tormented day and night for ever and ever (Latin: saecula saeculorum)," it is not to be taken in such a way that no sentence of God’s mercy is left at all: but because very long times (Latin: saecula saeculorum) are appointed for punishments, which human frailty cannot measure. It can happen that even the evil spirits themselves, who are now obstinate in their wickedness, after very long times are freed by God’s mercy, when they put off the nature which they corrupted, and are reformed to that for which they were created. But if this is so, it does not detract from divine justice, because they will have paid for a very long time what they deserved.
Source: Ambrosius Autpertus--Commentary on the Apocalypse (c. 784 AD), Book 10, ch. 36. Patrologia Latina 89:1391.
So Augustine and those who follow his reasoning are clearly wrong when they say that the Greek and Latin words often translated eternal punishment in Matthew 25:46 must have that meaning or we're left with no promise of eternal life. Matthew 25:46 can be translated "And these will go to the chastening of that Age, but the just to the life of that Age” without robbing us of the promise of eternal life. And the Greek word translated "punishment" in Matthew 25:46 (kolasis) is a word that was originally used for pruning trees. Which in itself suggests that the purpose of punishment is remedial (and that therefore the duration of the punishment cannot be endless.)