# Ultrafilters and principal filters [closed]

Can someone give me an example of an ultrafilter which is not principal?

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## closed as off topic by HJRW, Mark Sapir, Simon Thomas, Andreas Blass, Gerald EdgarSep 27 '11 at 14:50

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Do you mean on a set or on a Boolean algebra? You need some fragment of the axiom of choice to construct a non-principal ultrafilter on the natural numbers, for example. (See the link in Igor Rivin's answer.)

However, the Boolean algebra consisting of finite and cofinite subsets of $\mathbb N$ has a non-principal ultrafilter, namely the collection of all cofinite sets.

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I should perhaps be mentioned that it's consistent with ZFC that there are definable non-principal ultrafilters on the natural numbers. For example, Gödel's axiom of constructibility (V=L) implies that there is a definable well-ordering of the reals. That suffices to produce a definable non-principal ultrafilter. In fact, V=L implies the existence of a $\Delta^1_2$ non-principal ultrafilter on the set of natural numbers. –  Andreas Blass Sep 27 '11 at 12:53