I'd like to explain why John Boehner has led the charge to shut down the government thanks to some combination of innumeracy and his lack of knowledge of game theory.
Some numbers to start us off:
The house has 433 congressional representatives in it.
193 are Democratic
240 are Republican
of the Republicans,
66 belong to the Tea Party Caucus.
By the magic of subtraction, we can see that
240 - 66 = 174 Republicans are NOT member of the Tea Party Caucus.
John Boehner, due to innumeracy or just plain not giving a damn-ness, has said that he won't bring to a vote any budget that won't get enough votes to pass the house solely with Republican support. For a bill to pass the house, it needs ceiling(433/2)=217 votes. This means that he won't bring a budget to the floor unless it can garner 217 Republican votes, rather than 217 votes total.
To get that many Republican votes, if we assume that all non-Tea-Party members of the Republican party care even slightly more about the continued operation of the US government than they do about Obamacare and governments invading ladyparts, he has to convince 217 - 174 = 43 members of the Tea Party Caucus to do something other than shut down the government. However, the Tea Party Caucus has, as its main planks, "No Obamacare", "Government Bad", and "Government Must Forcibly Interfere With A Lady's Medical Choices". Three things that Obama will simply not sign into law.
Because Boehner refuses to consider Democratic votes as "votes" that could be used to pass a budget, the Tea Party controls congress. As long as the Tea Party controls congress, nothing is going to get fixed, because the Tea Party's official position on compromise (inasmuch as there is such a thing as the official position of the Tea Party besides "the future is bad and scary") is that compromise is the same as losing.
The government is going to stay shut down. We're going to hit the debt ceiling and the full faith and credit of the United States will be called into question. This will happen because John Boehner has publicly promised that, in the house, only Republican votes count.
John Boehner is a terrible human being who is bad at his job and can't add, and his terribleness is going to bring us all down.