Recently I learned that it can't be proved that mathematical axioms are consistent. And furthermore, in 1900s math was based on an inconsistent system of axioms.

So, are there any results, that were true in old axioms, were believed to be true (so obvious paradoxes don't count), but are false in modern axioms?

If there are some, then does it mean I can be skeptical about any proved statements, just because we can never be sure that our axioms are consistent?

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    $\begingroup$ Where did you learn that? $\endgroup$ – José Carlos Santos Apr 2 at 9:10
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    $\begingroup$ The goal of the modern axiom systems was to get a solid footing under the math that we had already done. Which is to say, the axioms were constructed with the explicit goal of changing as few results as possible. I'm not saying there aren't any, but I think you will have to look hard to find them. $\endgroup$ – Arthur Apr 2 at 9:10
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    $\begingroup$ One particular bit of mathematics, namely set theory, was based on assumptions that we now know to be inconsistent from its beginning in the late 1800s until about the first decade of the 1900s. But that does not infect all of the rest of mathematics, which was at that time done in the same way it had always been, without considering it to be applications of set theory, like some modern descriptions are wont to. $\endgroup$ – Henning Makholm Apr 2 at 9:13
  • $\begingroup$ @JoséCarlosSantos AFAIK Russel's paradox lead to creation of ZFC, because it has shown inconsistency of axioms that were before ZFC $\endgroup$ – Arqwer Apr 2 at 9:16
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    $\begingroup$ There were no set theory axioms before ZFC. $\endgroup$ – José Carlos Santos Apr 2 at 9:37

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