Libertarianism accepts that if everyone acts in their own selfish interests, society does not collapse. On the contrary, selfish people benefit others whenever they engage in mutually voluntary trade. Greater net wealth is created because of a lack of disincentives caused by the socialist principle of rewarding need. A libertarian society will be a very rich one.
Unfortunately it will also be a very fragile one, which would collapse as quickly as every attempt at communism has collapsed - because of the tragedy of the commons in defending against invasion.
A large hostile group (one not based on libertarianism) will invade the libertarian one. Clearly they would want to invade because the libertarian society would be much richer, and have a great deal of wealth to steal. While the libertarians would be free to join (or even wealthy enough to employ) an army to repel the invaders, individuals will choose not to do so.
The apocryphal shepherd puts his sheep on the commons knowing those commons will be grazed out and washed away - but he still puts his sheep onto it because any sacrifice of his is so unlikely to change the outcome. So too the libertarian doesn't bother to fund the defense force because his individual sacrifice will not repel the invaders.
While some libertarians argue that it would be in the interests of the libertarians to fund a defense force voluntarily, they are making the same argument that socialists make when they argue that peasants will work hard on communal farms because their actions feed everyone. Their argument is clearly naive and wrong.
The libertarian who believes that voluntary defense contributions will fund an effective defense force to protect his libertarian society may be more economically astute than than his socialist counterpart, but he is just as naive about human nature.
The only effective way to maintain a libertarian system is by collective means - forcing people to pay taxes to toward defending that system.
Just as pure collectivism (in the form of socialism or communism) collapses, so does pure individuality (libertarianism). The difference is that communism collapses from within, whereas libertarianism will collapse from external attack.
If all the world were libertarian, this would not be a problem, however with the plethora of popular ideologies in the world which are based on controlling others, universal acceptance of libertarianism is unlikely to happen soon.
In the meantime libertarians will have suffer the apparent contradiction of forcing people to contribute towards the most potent symbols of collectivism - the armed forces which have traditionally been used to subjugate individuality.
The only legitimate government function is to maximize individual freedom.