Are there still any paradoxes in modern math?

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if I google for paradoxes in math, all I find are ancient paradoxes which already have a hack or solution how to merge them out. Now I'm wondering if there are still any paradoxes in modern math, where nobody has found a way to eliminate or explain them. An example for a solved paradox would be the tortois paradox, which can be explained with infinite series.

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I assume you mean contradiction by the term paradox. Modern mathematics is based on formal axiomatic method. Particularly ordinary mathematics can be derived from ZFC set theory. No such paradox so far has been derived within ZFC. By Gödel's Incompleteness Theorems, in fact it is not possible to know within ZFC whether or not ZFC contains any contradictions. This does not mean that ZFC is inconsistent. It just means that "if it is consistent, it cannot be proved within ZFC".