So if you google “impossible to pay off the US national debt” you’ll find an article that’s been circulating on many conservative blog sites that argues why the central banking + fractional reserve banking model is broken and that it is mathematically impossible for the US government to pay off its debt. 
I actually stumbled on that article a long time ago and it seemed to pose a very interesting paradox, and it’s been turning around in my head for the last year or so because I couldn’t find any good resources online that give a direct refutation or clarification of the phenomenon it describes. Finally today, after listening to an NPR Planet Money report on the debate between dollar bills and dollar coins and the key idea of seigniorage, I think I understand what’s going on. It’s not terribly difficult, but I guess no one else thought it was interesting enough to write down (or Google didn’t care to list such rebuttals as high as the original article in the search results).
The basic paradox.
The (most basic version of the) paradox, in a nutshell, is that the US government as of March 2012 has about $15.6 trillion in debt, while the money supply (let’s say M2) is roughly $10 trillion. (The exact numbers don’t really matter, just the fact that the outstanding debt is larger than the money supply.) Ergo, it is mathematically impossible for the US government to pay off its debts, because even if it were to collect all of the money in circulation, it would not be enough to pay off the debt. QED, right?
-  As far as I can tell the article is of unclear authorship and it is reposted without attribution across many different sites, so let me not link to any particular blog and rather suggest you go and find it yourself on Google. If someone does know the authorship of this article please let me know and I’ll cite it appropriately. ↩