3. Ordinal numbers.
Where cardinal numbers describe 'size' by setting up a pairing between sets, ordinal numbers are concerned with order (first, second, third, etc).
So, the smallest ordinal number just has one thing (technically,, we could start with 0 things, but this is to make the whole ideas easier).
The second ordinal number has two things, with one smaller than the other. And example 1<2, so the set {1,2} with this ordering represents the second ordinal.
Now, when pairing sets for ordinals, we have to make sure the pairing preserves the ordering. So, an *order* pairing of {1,2,3} and {1,3,5}
would have to send 1<->1, 2<->3, 3<->5.
The difference is that for cardinals, we could have the pairing 1<->3, 2<->5, 3<->1. This is not allowed for ordinal comparisons.
Now, to add two ordinals, we put the first one in front of the second one. So, with {1<2<3} and {4<5<6}, to add them we have {1<2<3<4<5<6}, which is the sixth ordinal number
(so we write 3+3=6). The other way around would have {4<5<6<1<2<3} which has the same *order* structure as the previous e ordering is screwed up from the 'normal' one.
So, now, let's look at an infinite ordinal. The one we choose is the set of natural numbers again, {1<2<3<4<5<...}. This is usually denoted by the Greek letter ⍵. To add it to an ordinal representing 1, we use a symbol not in the naturals, say 1~{x}.
So, now, 1+⍵ is the set {x<1<2<3<4<..}. With a bit of thought, we can see that a pairing with x<->1, 1<->2, 2<->3, etc, will pair off 1+⍵ with ⍵, so we get
1+⍵=⍵.
Good enough.
But what if we do it the other way around? Now, ⍵+1 corresponds to a set like {1<2<3<.....<x}. Here, all the natural numbers are smaller than the x! Since x is now a largest member of this set, and since ⍵ does NOT have a largest member, the two are different!
So, for ordinal numbers, 1+⍵ is not the same as ⍵+1. When looking at order properties, order matters.
Now, to add ⍵+⍵, we have to put one copy of ⍵ after another copy! To distinguish them, I will use {1'<2"<3"<..}
for the second. Then we see that ⍵+⍵ corresponds to the set
{1<2<3<....<1'<2'<3'<....}.
This is different in order structure than ⍵.
Hence, for ordinal numbers, when we add two infinite ordinals we do NOT get the same ordinal back.
Multiplication is a much trickier thing. But again, the order matters and 2*⍵=⍵ is different than ⍵*2, which is the same as ⍵+⍵.
I hope this helped!
Where cardinal numbers describe 'size' by setting up a pairing between sets, ordinal numbers are concerned with order (first, second, third, etc).
So, the smallest ordinal number just has one thing (technically,, we could start with 0 things, but this is to make the whole ideas easier).
The second ordinal number has two things, with one smaller than the other. And example 1<2, so the set {1,2} with this ordering represents the second ordinal.
Now, when pairing sets for ordinals, we have to make sure the pairing preserves the ordering. So, an *order* pairing of {1,2,3} and {1,3,5}
would have to send 1<->1, 2<->3, 3<->5.
The difference is that for cardinals, we could have the pairing 1<->3, 2<->5, 3<->1. This is not allowed for ordinal comparisons.
Now, to add two ordinals, we put the first one in front of the second one. So, with {1<2<3} and {4<5<6}, to add them we have {1<2<3<4<5<6}, which is the sixth ordinal number
(so we write 3+3=6). The other way around would have {4<5<6<1<2<3} which has the same *order* structure as the previous e ordering is screwed up from the 'normal' one.
So, now, let's look at an infinite ordinal. The one we choose is the set of natural numbers again, {1<2<3<4<5<...}. This is usually denoted by the Greek letter ⍵. To add it to an ordinal representing 1, we use a symbol not in the naturals, say 1~{x}.
So, now, 1+⍵ is the set {x<1<2<3<4<..}. With a bit of thought, we can see that a pairing with x<->1, 1<->2, 2<->3, etc, will pair off 1+⍵ with ⍵, so we get
1+⍵=⍵.
Good enough.
But what if we do it the other way around? Now, ⍵+1 corresponds to a set like {1<2<3<.....<x}. Here, all the natural numbers are smaller than the x! Since x is now a largest member of this set, and since ⍵ does NOT have a largest member, the two are different!
So, for ordinal numbers, 1+⍵ is not the same as ⍵+1. When looking at order properties, order matters.
Now, to add ⍵+⍵, we have to put one copy of ⍵ after another copy! To distinguish them, I will use {1'<2"<3"<..}
for the second. Then we see that ⍵+⍵ corresponds to the set
{1<2<3<....<1'<2'<3'<....}.
This is different in order structure than ⍵.
Hence, for ordinal numbers, when we add two infinite ordinals we do NOT get the same ordinal back.
Multiplication is a much trickier thing. But again, the order matters and 2*⍵=⍵ is different than ⍵*2, which is the same as ⍵+⍵.
I hope this helped!
