what do areas of math like Set Theory bring us?
In: Mathematics
Set theory, like all mathematics, is a way to compress our intuitive notions into symbols, and define rules to manipulate these symbols. This then allows us to apply these rules to statements that are too complex for our intuition, and still come to conclusions that we can be sure are correct.
Example for set theory: you intuitively know what to make of sentences like “Socrates is a Greek” or “all Greeks are humans”, and that these sentences leave no doubt about the fact that Socrates is a human. You don’t need mathematics for that. But you can create a way to translate these ideas into mathematical notation (“Socrates is an element of the set of Greeks”), and define axioms and rules, and test that the results from applying these rules conform to what you intuitively know to be true.
Then you can use these rules to tackle a bigger problem. Say you want to draw a square where the side length is a whole number of inches, such that the diagonal is also a whole number of inches. How large does the square have to be, and is this even possible? It’s not a question where the answer is obvious. If you just try around, you will find the square with the side length of 70 where the diagonal is close enough to 99 that you might think you have found the solution. But applying mathematical rules (among them some set theory, namely that a number cannot at the same time *be* and *not be* in the set of even numbers) you can show that it is a futile exercise, and such a number does not exist.
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