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One of the most well-known examples is Frege's axiom of general comprehension, which asserts that for any definite property $P$, one may form the set $$\{x\mid\ P(x)\ \},$$ consisting of all objects $x$ with property $P$. This axiom formed an important part of the basis of Frege's Begriffsschrift. But it was famously refuted by the Russell paradox, which shows that there can be no set $R=\{x\mid\ x\notin x\ \}$, consisting of the sets $x$ that are not members of themselves, since then $R\in R\iff R\notin R$, a contradiction.