I was playing through Portal 2 some weeks ago and came across this awesome little math-related Easter egg. Close to the end of the game you run across this sign about paradoxes and the last statement about sets in the box of quotations is part of the paradox surrounding the universal set.
Let me make my Foundations of Mathematics professor proud and try to explain this contradiction. Firstly, a set is a collection of things called elements, anything you like, actual things like books, or concepts like numbers, and those collections usually contain things that are related like the set of My Favorite Books, or the set of whole numbers, but they don't necessarily have to have a relationship. Sets can even be comprised of other sets, for example I could compose a set of My Favorite books, My Favorite Movies, My Favorite Shapes, etc. Secondly, the universal set, as the name suggests, is the set that contains everything, all the elements individually, and all the possible sets of elements (similarly, if I recall correctly, the term can be applied to a limited point of view, perhaps just all books if that is what you're interested in looking at). The paradox comes from the universal set being the 'set of all sets.' If the universal set contains everything then it must contain itself because the universal set is part of everything that exists. However, if it did contain itself then there would be a bigger set containing everything and the universal set and hence that would be the universal set and the set we just included couldn't be the universal set. With this new universal set, we could once again included itself, making an even bigger set, then that would have to be the universal set, etc. etc. ad infinitum. A similar effect can be seen with the liar paradox--the first statement in the box. I hope this explanation is correct. In any case, there is a ton of literature about these concepts online if you're looking for more info :D
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