Okay, well aiwn meaning depends on context. Plural is almost always 'forever', like in Rom 1:25. Singular usually means a dispensational period (though remember Dispensations in Bible aren't like the 'scholars', but the 2100's). So there's that to examine.
More importantly, Christ could pay for all sins on the Cross even if they never cease, since in hell folks keep on sinning, and so too in the Lake of Fire. In finance, we call that an 'annuity' where you pay a single up-front lump sum that keeps on generating INCOME, which then PAYS forever. THAT is what I think the Cross accomplished. So sin CAN go on forever.
That's what I do for a living, calculate what DEPOSITS need to be made for ONGOING INCOME in pension plans. For humans, it's a finite calculation; but a deposit invested that never is touched (principal is never reduced) -- can generate income forever.
For there's also the opportunity cost (economic term) of all the SHORTFALL in Romans 3:23. Part of the cost of sin, is the LIMITATION on soul growth, which of course lasts forever, too.
So God builds FACTORIES from His Son; for yes, the total number of SOULS (angelic and human) are FINITE. We are DIVIDENDS of Him, so the noble deposit (Thieme's translation of 2Tim1:14, meaning MD) can go into our heads.. now, and in eternity. So think: a deposit INVESTED, yields income. Christ paid the WHOLE THING, and as a result we can even GET His Thinking. Which, initially are DIVIDENDS INVESTED, resulting in more income.. forever. Just like Christ said in John 19:30, tetelestai -- as you'll hear Thieme translate, 'Finished in the past with results that go on forever', aka perfect tense.
So the whole point of living post-salvation, is to get that 'noble deposit' (kalos paratheke, 2Tim1:14, Thieme translation), of His Thinking, which produces mini- 'Christ in you, the confidence of glory.' All life is about getting saved then getting Doctrine to live in your head, X+Y+Z, so in Z, we are 'factories' of pleasing 'smell' of His Thinking, real worship, Romans 12:1's logiken latreian. THINKING, not body stuff. Wrong thinking=wrong action, anyhow.
In finance, it's called 'present value'. A sum NOW which will be INVESTED and as a result will keep on growing and produce income. When you buy a 'lump sum annuity', that's why what you pay is so much less than what it will produce: the insurance company will be making money ON that lump sum you pay, enough to profit and pay you the annuity, too. That's what I think happened at the Cross, because of the terminology in Isaiah 53:12 'wa yehalleq shalal' (sorry, I can't type with Hebrew letters on this computer).
We can see the proof of this in our own lives, even with a simple Faith Rest. In the past, some Bible verse 'got deposited' in your soul, you believed it; later on, you recall it over and over and LIVE ON its promise. The deposit, might have only occurred once, but the 'income' occurs every time you recall (John 14:26) and USE it to keep going. Isaiah 54:1, Neh 8:10, etc.
Else the whole Bible makes no sense, for God must be paid for all sin, and that means the entire cost of creation, in heaven or hell, including angels (fallen angels still sin), etc.