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Table Of Contents
Mathematics of Machine Learning
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Let’s return to a remark I made earlier: naively defining sets as collections of things is not going to cut it. In the following, we are going to see why. Prepare for some mind-twisting mathematics.
Here’s a riddle. A barber is “one who shaves all those, and those only, who do not shave themselves.” Does the barber shave themself? There’s no good answer: either yes or no, the definition implies otherwise. This is known as the barber’s paradox. It’s more than a cute little story; it’s a paradox that shook the foundations of mathematics.
As we have seen, sets can be made of sets. For instance, {ℕ,ℤ,ℝ} is a collection of the most commonly used number sets. We might as well define the set of all sets, which we’ll denote with Ω.
With that, we can use the set-builder notation to describe the following collection of sets:
In plain English, S is a collection of sets...