what do areas of math like Set Theory bring us?
In: Mathematics
This is the core theories behind computer scienc, encryption theory, probability, and lots of other very important sciences. For example the theory behind relational databases are as the name suggest founded purely in set theory.
What do you mean by “like” set theory? Set theory is towards the extreme “pure” end of the pure-applied spectrum and doesn’t really have much in the way of direct practical applications, but it’s widely used as a foundation to help understand other areas of maths – part of the motivation for studying set theory is that sets are so general that you can define most other mathematical concepts *as* sets. Set theory notation is also very convenient and is used all over maths and related subjects, particularly the set membership symbol ∈, the union and intersection symbols ∪ and ∩, and set-builder notation.
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.
Set theory is the first common foundation of math.
Before set theory, there are no common foundation. Arithmetic and number theory follow their own rule. Analysis follow their own rule. Geometry follow their own rule. There are no consistency, there is no guarantee that they’re compatible with each other.
Set theory allow you to do that. When you use set theory as foundation, everything is made of set, and follow the same axioms. And your mathematical objects can be defined explicitly, for example “a natural number is a set that satisfies property …”.
After set theory, other foundations had been suggested, which had improvement over set theory. For example, they might be easier for computational purposes, or more suitable for computer proof-checking, and these had uses in bug-free programming.
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