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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 natural-seeming principle is one of the main axioms of what is now known as naive set theory, and formed a central axiom in Frege's Begriffsschrift and later the Grundgezetze (which I see now that you said you weren't interested in, oh well), intended as a formal logical foundation of arithmetic and all mathematics. But the axiom was famously refuted by Betrand Russell with the Russell paradox, showing 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.