>>11668298this is called "Russel's Paradox". You can find the contradiction on Wikipedia. Long ago, matbematicans used Naive Set Theory, which was a theory that ended up not being consistent due to this paradox. Modern mathematicians use a theory called ZFC which is much much more restricted and seems to be consistent as far as we know, but it can never be shown due to Gödel's second incompleteness theorem.
here's a TL;DR: there are many many different theories on what it means to be a set. Naive set theory was kind of the first, but also ended as a failure because it was too strong for its own good.
you can rank the "strength" of a set theory by how big sets are allowed to get. Naive set theory allowed you to make a set of all sets, which is much too big of a set and destroyed the theory. ZFC and other set theories limit you to smaller sets that will not cause contradictions.
it's essentially the case that in set theories, if theory A lets you create a massive set that is too large to be created in theory B, then theory A can prove more things(but you run the chance of it being inconsistent)
