[Copernicus]

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the main purpose of public education is to give every child the opportunity to get an education. The system often fails but in my opinion it is better thant no trying.
No public school-> poor children have no opportunity to sucess in life -> a free society that is not so free.

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Why do you assume that there would be no cheap schools?
It's not like public schools are a free lunch.

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Because of the free rider problem, as is often the case. Property taxes that pay for most public education are payed by users and non-users alike (which is appropriate because there are signficant social benefits from broad based educational opportunities). Give the non-users a choice to voluntarily contribute or not, and everyone that opts not to increases the burden on users.