Suppose I created a security that would give me back all the loose change found on the streets of Manhattan at any one instant. How much is that worth - $5, $20, more? (there's a lot of streets, and I'm always surprised how often I find a coin).
But I can guarantee that it's worth at least $1.00. That would be a AAA risk. And Moody's would rate it so, just by looking at the statistical properties of coins * streets.
This adds 1 security to the totals. The 'new idea' is that investors were willing to invest in new 'slices' of existing pools - chosen so that they'd get a AAA rating.
The idea itself isn't bogus. But the historical statistics of the subprime market were not valid for the wave of 'new money' moving into the Florida & Vegas markets... Subprime mortgages were a different animal than the rating agencies modeled, and no-one had any incentive to call them out on it.
No, but there is a theorem -- or call it an axiom -- that the amount of debt issued is equal to the amount of debt owed... okay, not a theorem, it depends on people being honest. Now the thing that's hard to believe is that we went from a situation in which the vast majority of debtors were unreliable to the current situation where a significant proportion are considered reliable, and did all of this without any really significant or obvious cultural change, and during a period in which bankruptcy was just as common as before... basically, the "mathematically impossible" thing is that there is a lot more debt being considered reliable but there aren't any more reliable debtors.
Now, you could contest the latter: are there more reliable debtors today? It's not impossible, but I don't see very many, beyond the US Treasury.