Now, we also want to be able to pair sets together, to take unions, and to take power sets (the powerset of a set A is the set whose elements are all the subsets of A). We also want to be able to say whether two sets are equal, so we’ll use the obvious definition that two sets are equal if and only if any member of one set is a member of the other set. For starters, we won’t let sets be members of themselves. So to avoid that messiness, we really will need a few axioms to specify when a collection of objects is a ‘set’. But does it belong to itself? I’ll let you work that out. OK, this is a collection of things so it’s a set. To see where the issue arises, try to define “the collection of all sets that don’t contain themselves”. So why do we run into an issue with sets? For example, consider the following python code:Ī is an array that contains the elements 1, 2, and a. This might seem a bit unintuitive at first, but we can define things that contain themselves all the time in programming languages. Furthermore, under this definition there’s nothing stopping a set from containing itself. So then a set is a thing, which means sets can contain other sets. Russell’s paradox goes as follows: we say that sets are just things that contain other things. But as Bertrand Russel pointed out, we’d end up with a nasty paradox, and mathematicians tend to dislike those. Hold up a second, why do I need rules? Why can’t we just say “A set is a collection of objects” and be done with it? Then I’ll want to lay down some rules that these things should follow. Now suppose I want to think about sets a bit more formally. Just now, I’m drinking a collection of atoms (mostly hydrogen and oxygen) from a cup-shaped object on my table. Sets, or collections of things, are a cornerstone of life. This blog post is about the kind of math that even pure mathematicians sometimes criticize as being too abstract to ever be useful. This blog post is not about that kind of math. Rather, mathematics is about finding relationships between things and pushing ideas to their theoretical limits, and every once and a while it just so happens that the things you discover happen to describe something in physics, or biology or chemistry or economics or some other field that would at first glance appear to be far flung from the rigor of math. Yes, some parts of mathematics allow us to describe and model natural phenomena, but this is hardly due to the nature of math. Many people have a peculiar notion that mathematics is meant to describe the real world, but this couldn’t be further from the truth. One asks “what if?” and then goes from there, following a path of reasoning that often passes through surreal and sometimes seemingly nonsensical territory to (hopefully) arrive at some deep theorem or insight into the nature of an object.
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