What is ∞ + ∞ = ????

[Succubus]

Here's a real number to fuck your head with.

A Tower of Powers.

...The pattern we’ve seen is each new level bundles a string of the previous level together by using a b term as the length of the string. For example:...

Well yes, everybody knows that, ffs!
But to be serious, I'm sure all contributors  to this thread are familiar with Graham's Number, but what fascinates me is that it's beyond any sort of visual analogy. I'm aware of the Plank cube approach but that's a bit 'mathy' for me. What I can visualise is, the volume of the 90 billion light year diameter universe couldn't hold enough ink to print that number.
Well shit in my mouth and call me bad breath! You can not be fucking serious!

I had to scrap my plan to lay off the weed for a bit.

[Haipule]

If you add ∞ to ∞, then what do you get?...

Eyeglasses + boobs = Jack

[polymath257]

If you add ∞ to ∞, then what do you get?...

The quick answer is that it depends.



There are several different types of infinity in mathematics (and I won't even go into philosophy).

1. Limits

Limits are ways of describing what happens to a variable when another variable gets closer and closer to a value without getting there, or when the second variable gets larger and larger.

So, for example, if you let y=(x+1)/x, as x gets larger and larger, y gets closer and closer to 1. We say that the limit of y as x approaches infinity is 1.

Now, sometimes the second variable gets larger and larger. For example, as x gets close to 0, y= 1/x^2 gets larger and larger. In this case, we say that the limit
of y as x approaches 0 is infinite.

Now, suppose that the limit of y as x approaches a number is infinite and, similarly, the limit of z as x approaches that value is also infinite. Then the limit of ywill also be infinite.

So, in this case, ∞ + ∞=∞.

But, it is crucial to remember that ∞ is not a number in this. It is simply a short-hand for saying that something gets larger and larger without bound.

2. Cardinality (sizes of sets)

A set is a collection of mathematical objects. Anything in the collection is called a member of that set. We describe sets by listing the members, separated by commas, between braces. So, A={1,2,3,4} is a set with members 1 and 2 and 3 and 4.

We say two sets have the same cardinality if they can be paired off in some way, with each member of one being paired with exactly one member of the other.
For example, the sets A={1,2,3,4} and B={5,7,9,11} have the same cardinality because we can pair 1<->5, 2<->7, 3<->9, 4<->11. This is not the only way to pair.
But we only need one way to say say the sets are the same cardinality.

Intuitively, cardinality corresponds to the size of the set, so sets with the same cardinality are of the same size.

Now, a natural number is an ordinary counting number, like 1,2,3,4,5,6,... We can look at the set of ALL natural numbers,
N={1,2,3,4,5,6.....}.

But this set has a strange property: we can pair it off with the (smaller?) subset of even natural numbers E={2,4,6,8,10,12,...} as follows:
1<->2
2<->4
3<->6
4<->8
etc.

In this, every natural number on the left is paired with exactly one even number on the right. So, in this formulation the set of natural numbers has the 'same cardinality'
as the set of even numbers, even though the set of even numbers is 'smaller' in a very real sense.

This property of having the same cardinality as a proper subset is characteristic of infinite sets. And the set of natural numbers is an infinite set.

There are many other sets that have the same cardinality as the set of natural numbers: the set of integers (both positive and negative counting numbers), the set of all fractions, and many others.

But, it turns out that if you look at the set of *all* decimal numbers, that is a larger cardinality! if you allow for infinite, non-repeating decimal numbers (like pi), there is no way to make a pairing to the natural numbers!

There is more than one 'size' of infinity!

None the less, if you take two infinite sets and 'add them' together, the result is another infinite set (of the larger cardinality), so in this sense ∞ + ∞=∞, although
we can do a more refined formula detailing the cardinalities.

3. I'll save ordinals for another post.

[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!

[Fireball]

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.

[Whateverist]

Actually I don't think there is any time limit on weaseling editing now .. not that I would know anything about that.  

[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.

[Whateverist]

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.

Truly weird.  I just went back to the 26th post in this thread which I wrote on Dec 2.  I was able to edit it and made a note to that effect.  The edit button should be on the bottom of your post, just left of "Give Kudos".

But don't sweat the small stuff.  I'm sure what you've already said will suffice.

[vulcanlogician]

No, dude. You must be special. The rest of us have a 1 hour time limit.

[Jackalope]

It's probably an incorrect permission setting. We'll look into it.

FWIW the hour limit is intended to apply to everyone except staff.