Sets and Terminology

Russell's Paradox

What do you want to contradict?

What do we usually contradict?

We want to contradict an object.

What kind of object do we want to contradict?

What is the theory named at the beginning of this?

We named set theory. Now what object do we want to contradict?

We want a set with a contradiction. Let's try to think of one...

What's something important to sets?

Think of any set... When you're done, continue and choose one that's most similar to the set you're thinking of.

What's different from the others?

What is another set?

What's the difference between $\{\}$ and $\{1\}$?

The empty set has nothing in it.

How would you be a bit more rigorous with this?

For all $x$, $x$ is not in the empty set.

What's the key word?

in

Should we come up with a convention for it?

We write "in" as $\in$.

$\{1,2,3,\dots,n-1,n\}$

@ What's special about this set?

Why didn't you include 0?

No reason? What did you say?

0 is not in the set.

How would you be a bit more rigorous with this?

0 is not in this set.

What's the key word?

in

Should we come up with a convention for it?

We write "in" as $\in$.

$\{0,2,4,\dots,2n-2,2n\}$

@ What's special about the set?

Why did you skip 1?

How do you relate 1 and this set?

1 is not in the set.

What's the key word?

in

Should we come up with a convention for it?

We write "in" as $\in$.

set of natural numbers

@ Welcome to the short section looking at the natural numbers. First, continue if you recall the controversy. If you're thinking of another property that makes the natural numbers special, you can skip the first few pauses.

The controversy is regarding $0$ and $0^0$.

How would you write the question of this controversy more rigorously?

Should we include 0 in the set of natural numbers?

That should give you another bit of background for the key word we're to look at later on.

What's special about this set?

Is it infinite?

It's infinite. How do we show that? Click continue after you've been thinking for a while or you think you've got it.

What is something in this set?

Assume you said $n$. Why isn't the set just $\{0,1,2,\dots,n-1,n\}$?

Think simply. If you come up with some complicated expression, why would it usually not work?

We haven't even defined a bunch of stuff! Also, you might skip over some natural numbers. What is the simple next number?

$n+1$ is also in the set

What is the word that keeps on appearing?

in

Should we come up with a convention for it?

"In" is written as $\in$.

set with everything (universal set)

@ What's special about this set?

Everything's included.

Write it more rigorously.

For all $x$, $x$ is in the universal set.

What do you think is the key word?

in

Should we come up with a convention for it?

We write "in" as $\in$.

We must focus on what's in the set. "In" is the essence of set theory. What was our goal again?

We wanted to construct a set that contradicts itself. What does it make sense to do?

Call this set $A$. Maybe you want to be a little more specific than saying that it contradicts itself...

How do we usually find contradictions?

We want $A$ to have some property and not have this property.

What property do you want to consider?

What has to do with in and sets?

We can consider if something is in something as a condition. Think about the contradictory property again...

It seems like a condition. How do we use it?

Define $A$ to be the set of $x$ such that $x\in x$. First, do you want to have a convention for defining sets?

$\{x|\phi(x)\}$ is the set of $x$ such that the property $\phi(x)$ holds. How do we write $A$ this way?

$A=\{x|x\in x\}$.

What's our goal?

Find a special property for $A$. What would be a nice property to consider?

Is $A\in A$?

Hmm, try it.

If $A\in A$, then what must be true?

$A\in A$...

Do you think we're going to get a contradiction without changing $A$?

Try it.

It's easier to change $A$. What else are we trying to do with $A$?

We're assuming that $A\in A$.

What do we want to get?

We want a contradiction.

What would a contradiction be?

We want $A\not\in A$.

How do we rewrite $A$ such that this is true?

$A=\{x|x\not\in x\}$.

Now is $A\not\in A$?

If we just say so by the previous part, then do we get a contradiction?

Do we just give up on this method?

No. We are determined to get a contradiction. If $A\not\in A$, then what must be true?

Think about the definition.

$A\in A$. Why are we happy?

This is a contradiction! Now people have motivation to wonder about what it means to be a set!

$A\in A$

@ What does this seem like?

It seems like something to consider. How does it inspire us?

What's our goal?

We want some property and the same property to be false. What can we set as this property?

We want $A\in A$ and $A\not\in A$.

What feels uncomfortable about $A$?

We haven't defined it yet. What would be a typical definition of a set?

What would it involve?

An element is in a set if it has some property.

What would be a good property?

Are there any simple properties that come up that aren't arbitrary?

Can you find a property that has occured really often up to this point in this article?

We consider whether something is in something! Do you want to a convention for this?

$\{x|\phi(x)\}$ is the set of $x$ such that the property $\phi(x)$ holds. How do we write $A$ this way?

$A=\{x|\_\_\_\in\_\_\_\}$. How should we fill in the blanks?

Do you want another variable?

Can we talk about $A$ before defining it?

$A=\{x|x\in x\}$. Let's investigate.

What is our source of inspiration?

We consider whether $A\in A$.

Should our inspiration be more important than what we define?

We may have to alter our definition. But anyways, if $A\in A$, then what happens by definition?

$A\in A$. What do we want?

We want a contradiction. Like what?

$A\not\in A$. What would be another definition for $A$?

$A=\{x|x\not\in x\}$.

What's the question you're about to ask?

Is $A\in A$?

By definition, $A\not\in A$.

How do we rewrite $A$ such that this is true?

$A=\{x|x\not\in x\}$.

Now is $A\not\in A$?

If we just say so by the previous part, then do we get a contradiction?

Do we just give up on this method?

No. We are determined to get a contradiction. If $A\not\in A$, then what must be true?

Think about the definition.

$A\in A$. Why are we happy?

This is a contradiction! Now people have motivation to wonder about what it means to be a set!