How to think of a set?

Question

I am doing self study for the last two months on functional analysis. As I get a bit used to the terms like space, topology, manifold, etc, etc, I realized that everything is defined in terms of set. If I look at the wikipedia page on set or set theory we can get quite a bit of explanation, but it did not clear me how to think of a 'set'. May be I am trying to think 'set' as some geometric object from which everything an be derived. May be I am wondering how a physicist would think of 'set'. I am worried that this question will be closed with lot of down vote but I still decided to try my luck. May be I asking few remarks from community how they think of 'set'. That might help me to develop some missing link, which I cannot explain properly.

Answer 1

The bad news

You're not alone in being confused about how to think of a set. Working out a sensible language for talking about sets cost mathematicians a great deal of sweat and tears. It's now considered one of the major achievements of 20th-century mathematics, and it's still being built upon today.

If you set out to explore our universe of sets, or the universes just beyond it, you'll quickly encounter sets that are dizzyingly, terrifyingly, hilariously counterintuitive. If you want my opinion, the best way to think of sets like these is don't.

The good news

On the other hand, if all you want to do is geometry, all the sets you meet will be tame enough that you can learn how to handle them just by looking at examples. In that spirit, let me give you some examples of sets that show up in geometry. It's often useful to have lots of examples, so this answer will be rather unfortunately long. Sorry!

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Any finite collection of things. The three musketeers are a finite collection of people, so intuitively they should form a set, which I'll call $\mathbf{M}$. In formal notation, we might define this set by writing $$\mathbf{M} = \{\text{Athos}, \text{Porthos}, \text{Aramis}\}.$$ The individual musketeers are called the elements of this set. Where a normal person would say "Porthos is one of the three musketeers," a mathematician might say, "Porthos is an element of $\mathbf{M}$," or write $\text{Porthos} \in \mathbf{M}$.

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The natural numbers. The numbers 1, 2, 3, 4, and so on are a collection of things, so intuitively they should form a set, which is usually called $\mathbb{N}$. Logicians, who need to use very formal language, talk about this set by giving a precise logical description of it. Fortunately, as a geometer, you can get away with an informal description like this: $$\mathbb{N} = \{1, 2, 3, 4, \ldots\}.$$ The dots are just another way of saying "and so on," in the hope that your reader will know what you mean.

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The natural numbers whose squares are less than 100. These numbers are a collection of things, so they form a set, which I'll call $\mathbf{S}$. We can produce this set by rummaging through the set $\mathbb{N}$ and grabbing all the elements whose squares are less than 100. This way of building sets is so useful that people made up a special way of writing it: $$\mathbf{S} = \{n \in \mathbb{N} \mid n^2 < 100\}.$$ This could be read out loud as, "$\mathbf{S}$ is the set of natural numbers $n$ with the property that $n^2 < 100$."

Because every element of $\mathbf{S}$ is also an element of $\mathbb{N}$, we say $\mathbf{S}$ is a subset of $\mathbb{N}$. In writing, $\mathbf{S} \subset \mathbb{N}$.

If you do some thinking, you can find a more concrete expression for the set $\mathbf{S}$: $$\mathbf{S} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}.$$

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The Euclidean plane. Imagine an infinite sheet of paper, perfectly flat and unmarked. Although it seems like a funny way of thinking at first, you can think of this sheet of paper as an infinite collection of locations, called points. This allows you to talk about the sheet of paper as a set, which I'll call $\mathbf{E}$.

One of the most interesting things you can do in the set $\mathbf{E}$ is measure distances. Given any two elements $p$ and $q$ of $\mathbf{E}$, I'll write the distance between them as $d(p, q)$.

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A circle in the plane. A circle can be described as a set of points in the plane that are all the same distance from a certain point $a$. The point $a$ is called the center of the circle, and the common distance is called the radius. The circle with center $a \in \mathbf{E}$ and radius $1$, which I'll call $C_1(a)$, can be described formally by writing $$C_1(a) = \{p \in \mathbf{E} \mid d(p, a) = 1\}.$$ Every circle is a subset of $\mathbf{E}$.

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A line in the plane. Pick two different points in the plane—that is, let's pick $a, b \in \mathbf{E}$ with $a \neq b$. Let's say $V(a, b)$ is the set of points which are equal distaces from $a$ and $b$. That is, $$V(a, b) = \{p \in \mathbf{E} \mid d(p, a) = d(p, b)\}.$$ If you think about it, you should be able to convince yourself that $L$ is a straight line!

Notice that many different choices of points $a$ and $b$ give the same line $V(a, b)$.

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A set of circles in the plane. Sets are things, so a collection of sets should form a set! Logicians have to be very careful about constructing sets of sets, but as a geometer, you shouldn't run into any problems.

Here's a description of the set of all circles with radius one: $$\{C_1(a) \mid a \in \mathbf{E}\}.$$ If we unpack the meaning of $C_a(1)$, our description of this set of circles expands to $$\{\{p \in \mathbf{E} \mid d(p, a) = 1\} \mid a \in \mathbf{E}\}.$$ This expanded description makes it clear that we're dealing with a set of sets.

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Another set of circles in the plane. Here's a description of the set of all circles whose radii are natural numbers: $$\{C_n(a) \mid a \in \mathbf{E}, n \in \mathbb{N}\}.$$

Every circle with radius one is a circle whose radius is a natural number, so $$\{C_1(a) \mid a \in \mathbf{E}\} \subset \{C_n(a) \mid a \in \mathbf{E}, n \in \mathbb{N}\}.$$

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The set of all lines in the plane. Let's say $\mathcal{L}_{\mathbf{E}}$ is the set of all lines in the plane. This set can be described by writing $$\mathcal{L}_{\mathbf{E}} = \{V(a, b) \mid a, b \in \mathbf{E}\}.$$ If we actually went through all pairs of points $a, b \in \mathbf{E}$ and drew the line $V(a, b)$ for each pair, we would draw each line many times over. That's okay: when you're describing a set using this notation, it's okay if you describe some elements more than once.

The set $\mathcal{L}_{\mathbf{E}}$ is useful because it tells you a lot about the geometry of the plane. In fact, even if you forget how to measure distances, you can do a lot of geometry just by playing with the sets $\mathbf{E}$ and $\mathcal{L}_{\mathbf{E}}$. For example, here's an expression for the set of lines that pass through a pair of points $a$ and $b$: $$\{L \in \mathcal{L}_{\mathbf{E}} \mid a \in L \text{ and } b \in L\}.$$ It's a very important fact about geometry that when $a \neq b$, this set always has exactly one element!

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The Fano plane. Once mathematicians got the hang of doing geometry by thinking of the plane as a set of points and then thinking of lines as a set of subsets of the plane, they realized that they might be able to create new kinds of geometry by starting with a set other than $\mathbf{E}$.

A cool example is the Fano plane, a set of seven "points" organized into "lines" of three points each. The relationship between the set of points $\mathbf{F}$ and the set of lines $\mathcal{L}_{\mathbf{F}}$ is very similar to the relationship between $\mathbf{E}$ and $\mathcal{L}_{\mathbf{E}}$. For instance, if you pick two points $a, b \in \mathbf{F}$ with $a \neq b$, you'll find that the set $$\{L \in \mathcal{L}_{\mathbf{F}} \mid a \in L \text{ and } b \in L\}$$ always has exactly one element. In the Fano plane, just like in the Euclidean plane, there's exactly one straight line through every two different points.

Answer 2

Unfortunately, "set" is as nebulous a concept as sets themselves. It is not a geometric object, and certainly not a physical object. It is instead an abstract concept that arose from the desire to specify what kind of collections can be described. Clearly the specification would be essentially specifying some kind of property that the elements in a collection must satisfy and that everything else does not satisfy. At the same time, certain mathematicians wanted the elements of a set themselves to be sets for whatever reasons that are clearly not related to the physical world. And so naive set theory was constructed, where it is allowed to create any set (of sets) that satisfy a property. Unfortunately, this is just an identification of sets with functions from sets to boolean (true or false), and anything of this sort is doomed to a contradiction in any reasonable formal system. Therefore Zermelo and Fraenkel came up with a set of axioms where the kind of set you are allowed to construct is severely restricted but there are other axioms make up for that. In this way the naive idea of having sets identified with their indicator functions is preserved, but at the cost of having some collections being proper classes instead of sets. There are other alternatives such as various type theories some of which more closely reflect the actual way we do mathematics than ZFC. But that is another topic.

And if you study first-order logic, you will realize that ZFC is a first-order theory where the domain is intended to consist only of sets, but that is when viewing it from the outside! Within the theory you cannot define the domain at all, and all you 'know' are the axioms of ZFC. Similarly, the basic 2-input predicate "$\in$" cannot be defined within ZFC, because you cannot express "$t \in u$" if you are not allowed to use any predicates at all! Thus if you do mathematics completely within ZFC both sets and "$\in$" will be undefinable. How then do we talk about ZFC itself? We would have to go outside it. If you notice, when studying logic itself we are using some kind of mathematics to state and prove theorems about logic, and that mathematics is usually ZFC!

This phenomenon is intrinsic to all languages, including formal languages like ZFC and natural languages like English, where some notions are not definable in the language. For example, one cannot understand the meaning of "if" without already understanding something equivalent to it. This is in fact related to why we need at least one inference rule in logic, usually modus ponens, because without rules we cannot do anything. But rules cannot be defined! Moreover, to use modus ponens we already need to understand "if", because the rule says "If you have derived formulas $α,α \rightarrow β$, then you can derive $β$.".

Answer 3

The easy answer, which probably isn't useful to you who do quite advanced courses, is that a set is a collection of objects. However the interesting part of sets are really the strength which this implies if we do interpretations of different sets.

Sets can not contain the same element twice, but we may identify $\{1,\{1\}\}$ with the ordered 2-tuple (1,1).

Functions may be simulated using sets by creating the graph, for instance $\{\{x,\{x+1\}\} : x \in Z\}$ could be identified with the function $f(x)=x+1$.

In the same sense we may define almost all mathematics by using such a simple humble concept. So to answer your question; I think about sets as the small very simple buildingblocks which we build mathematics from, and it all starts by just having a collection of things.

Answer 4

A set can be thought of as an inclusion rule, or "All the objects which satisfy a given inclusion rule."

"All odd numbers". If you're an odd number, you're in the set. If you're not, you're not in the set. Or, "All the states in the US whose names start with 'A'".

The set might be empty: "All odd multiples of 2". The empty set is actually important, but you'll learn that as you go. Just like zero is an extremely important and useful number but they don't teach you that at first.

You can even have a set defined like this: "The numbers 1, 2, and 5". Anything that is not one of those numbers is not in that set.

Answer 5

What really matters about a set, $S, $ is that for any element, $ x, $ in the 'universe' exactly one of $x \in S, $ $ x \notin S$ is true. You can put stuff inside of a set, thus distinghishing it from everything that isn't inside if the set. It's a symbolic way of talking about "this stuff here."

Don't confuse how to think about a set with how to define a set. What you think about a set controls how you use a set and how well you understand theorems that use sets. How you define a set is how you prove those theorems.

The important thing is, if a theorem conflicts with what you think about a set, then you have to change the way you think about a set. The definitions that describe how a set is used are what a set really is. those definitions should correspond to your intuitive idea of what a set is. But once you state a definition, then the set becomes the thing you defined; which may no longer be the thing you imagined.

Answer 6

Let me begin by saying that this is I think of sets. Hopefully it is helpful.

My intuition of what a set is, as vague as it is, is that it is a thing with stuff in it (or in the special case of an empty set, with no stuff) (more exactly, a set theory is a theory that has an epsilon relation). However, that is not quite enough to work with, so one naively begins to ask that we have various ways of building sets that reflect our activities. So one may want to be able to take the union of two sets, or more importantly, specify a set whose elements are all things that satisfy some property. More precisely, one may want to form the set, $$evenInt=\{x:x\mbox{ is an integer, and } x \mbox{ is even}\},$$ or other such things.

So one posits a universe (never mind where it or it's inhabitants came from), and we can form all sets that satisfy some property, or even perhaps form "random subsets". However, all is not well in this paradise since using this formulation, one may form the set, $$Russ = \{S: S\mbox{ is a set, and }S\notin S \}.$$ This looks like a perfectly good thing to do, until you ask the question "Does $Russ$ contain itself as an element"? Well if it does then it doesn't and if it doesn't then it does. This is called Russell's paradox. See http://en.wikipedia.org/wiki/Russell's_paradox .

So then people asked "How do we fix this"? An answer that was come up with is the ZFC axioms. These axioms appear to be consistent. Moreover, they seem to capture the types of constructions that people want to make on sets, without running afoul of Russell's paradox and other paradoxes.

This theory also has the effect that large chunks of mathematics is formalizable in this theory, which is another goal of set theory. One problem or feature with ZFC is that it treats everything as the same kind of thing, namely a set whose elements themselves sets, all the way down to the empty set (one might argue that certain type theories are set theories that do not have this property). See https://golem.ph.utexas.edu/category/2013/01/from_set_theory_to_type_theory.html form more on that.