I'm afraid that saying "just ask the PIIGS" is to miss the crucial point. Those countries are members of the Eurozone, and therefore their governments are currency users rather than issuers. Therefore, an analysis that draws upon your everyday household experience carries at least some water, even though it is entirely inapplicable to a monetarily sovereign government such as the US government.
Other than that, I can only recommend that you try to consider all the relevant dynamic effects in the macroeconomy. For example, if money would indeed become worthless, this would not happen overnight due to the immense inertia of an economy as large as the US economy. It would be a drawn out process.
Throughout that process, as a consequence of money losing value, the nominal GDP would increase, and therefore the debt-to-GDP ratio would decrease, which means that the system has a very strong self-stabilizing tendency.
Other than that, I can only recommend that you try to consider all the relevant dynamic effects in the macroeconomy. For example, if money would indeed become worthless, this would not happen overnight due to the immense inertia of an economy as large as the US economy. It would be a drawn out process.
Throughout that process, as a consequence of money losing value, the nominal GDP would increase, and therefore the debt-to-GDP ratio would decrease, which means that the system has a very strong self-stabilizing tendency.