What is ∞ + ∞ = ????

[polymath257]

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!

There are maybe 6 people here who are going to get this. I could be wrong. The guy who originally posed the question is a serial sock (5?) times that I know of, BTW.

As an aside, you have prime and double prime where you are adding ⍵+⍵, which may cause confusion. If I'm wrong, I'll edit my post after you do so that both are correct. I believe the edit window is 1 hour now, since some people like to weasel on their statements. Not accusing you of that, BTW.

Thank you for pointing out the typos. I no longer see an edit tab, so this will have to serve as a heads-up.