Explain to me how not raising the debt ceiling will cause us to default on our debts?

Isn't it true that people who claim this above statement to be true are disregarding that it's our spending that is what costs new debt?  Isn't a budget voted on each year (or two).  Isn't the real problem with not raising the debt ceiling that people won't actually vote to cut spending?

Doesn't cutting the debt ceiling force more attention on actually reducing the federal budget?

There has got to be something that I'm missing here.

Edited to replace 'extending' with raising.

We need to borrow money to pay the people we already owe money to.

They'd have to cut the federal budget in half to stop borrowing money and make the minimum monthly payment on the national credit card.

Because to get out of debt you must go deeper into debt, at least that's the way the government does it.

Unfortunately, this is the case.

 

We are paying our current debt obligations with borrowed money.  No more borrowed money means no money to pay current obligations.  That means default.