Ordinal arithmetic
Ordinal arithmetic is like counting, but instead of using numbers like 1, 2, 3, etc., we use special words to describe big groups of things. These words are called ordinals.
Let's start with some simple ordinals. The first ordinal is called "first." We use it when we're talking about the very first thing in a group. The second ordinal is called "second." We use it when we're talking about the second thing in a group. The third ordinal is called "third." We use it when we're talking about the third thing in a group.
Now, let's move on to some bigger ordinals. The fourth ordinal is called "fourth." We use it when we're talking about the fourth thing in a group. The fifth ordinal is called "fifth." We use it when we're talking about the fifth thing in a group. And so on.
But what happens when we have a really big group of things, like all the numbers between 1 and infinity? That's where ordinal arithmetic comes in.
Ordinal arithmetic helps us figure out how to count really big groups of things. For example, if we want to count all the numbers between 1 and infinity, we start by saying "first" for 1, "second" for 2, "third" for 3, and so on. But eventually, we run out of words for all the numbers between 1 and infinity!
That's why we need bigger ordinals. The first big ordinal is called "omega." We use it to describe all the numbers between 1 and infinity. So instead of saying "first," "second," "third," and so on, we say "omega," "omega + 1," "omega + 2," and so on.
But what about even bigger groups of things? That's where ordinal arithmetic gets really interesting. We can keep adding ordinals to make even bigger ones. For example, if we take two groups of things and put them together, we get a new group that's bigger than either of the original groups. We use a special operator called "plus" to combine two ordinals.
So if we take the group of all the numbers between 1 and infinity, which we call "omega," and we add it to itself, we get a new ordinal that's even bigger than omega. We call this new ordinal "2 times omega," or just "2 omega" for short.
We can keep adding ordinals like this to make even bigger ones. For example, we could add omega, 2 omega, and 3 omega together to get a new ordinal called "6 omega."
Overall, ordinal arithmetic is a way of counting really big groups of things using special words called ordinals. By combining ordinals using operators like "plus," we can create even bigger ordinals that help us understand the infinite complexity of the world around us.
Let's start with some simple ordinals. The first ordinal is called "first." We use it when we're talking about the very first thing in a group. The second ordinal is called "second." We use it when we're talking about the second thing in a group. The third ordinal is called "third." We use it when we're talking about the third thing in a group.
Now, let's move on to some bigger ordinals. The fourth ordinal is called "fourth." We use it when we're talking about the fourth thing in a group. The fifth ordinal is called "fifth." We use it when we're talking about the fifth thing in a group. And so on.
But what happens when we have a really big group of things, like all the numbers between 1 and infinity? That's where ordinal arithmetic comes in.
Ordinal arithmetic helps us figure out how to count really big groups of things. For example, if we want to count all the numbers between 1 and infinity, we start by saying "first" for 1, "second" for 2, "third" for 3, and so on. But eventually, we run out of words for all the numbers between 1 and infinity!
That's why we need bigger ordinals. The first big ordinal is called "omega." We use it to describe all the numbers between 1 and infinity. So instead of saying "first," "second," "third," and so on, we say "omega," "omega + 1," "omega + 2," and so on.
But what about even bigger groups of things? That's where ordinal arithmetic gets really interesting. We can keep adding ordinals to make even bigger ones. For example, if we take two groups of things and put them together, we get a new group that's bigger than either of the original groups. We use a special operator called "plus" to combine two ordinals.
So if we take the group of all the numbers between 1 and infinity, which we call "omega," and we add it to itself, we get a new ordinal that's even bigger than omega. We call this new ordinal "2 times omega," or just "2 omega" for short.
We can keep adding ordinals like this to make even bigger ones. For example, we could add omega, 2 omega, and 3 omega together to get a new ordinal called "6 omega."
Overall, ordinal arithmetic is a way of counting really big groups of things using special words called ordinals. By combining ordinals using operators like "plus," we can create even bigger ordinals that help us understand the infinite complexity of the world around us.