You need to pay taxes and have kids for the numbers to work out.
If your pension benefits are calculated based on a 4 workers/retiree ratio, but then your whole generation has like 1 kid per family then the system will obviously break down...
No, it would only imply that the worker/retiree income ratio (and tax/pension burden) are somewhat constant which is arguably the case; german pensions specifically rise with wages and get adjusted for inflation.
Productivity gains on the other hand get easily eaten up by increased consumption/expectations, or are overstated to begin with: producing 5 times more TVs/ipads does not make the plumber cheaper (nor a house), and unaffected professions actually suffer (=> baumol effect).