In mathematics, sets are simply collections of objects. Set theory is the branch of mathematics that studies these collections of objects.

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Classical theory is a reference to established theory. Fuzzy set theory is a reference to theories that are not widely accepted.

Set theory was founded by Georg Cantor in 1873.

The theory is evaluated in the context of the new information. The information may support the theory, it may require a tweak to the theory to accommodate the new information or it may require a major re-think of the theory.

No - you can use evidence from an experiment to support a theory, but they are not the same thing.

A theory is an idea as to why something is, how it works, etc. that is strongly supported by evidence.

An experiment is set up in order to gather information to support a theory or to help a hypothesis become a theory.