[This is basically mathematical brain dump, and the only real editing was to remove sections that got duplicated in the dumping process. Read at your peril.]
In high school I came up with my own definition of infinity:
Infinity is the point at which things stop making sense.
Throughout university I stuck to this definition.
I've come to realize that many (though by no means all) of the oddities I have long associated with infinity are actually properties of gaps. This is because the infinity that we almost always talked about in the branches of math I partake in was sideways eight infinity (∞) which is in fact the gap at infinity.
We occasionally touched on things like the alephs (the ℵ numbers) which are cardinals, but my chosen path through math seldom dealt with cardinal numbers (and basically never touched ordinals) and instead "infinity" almost always was ∞ in our discussions.
So, here's the thing, ∞ is not a number (which we always acknowledged by saying: "infinity is not a number") it's a gap. A lot of the weirdness of ∞ (for example the fact that ∞ + 1 = ∞, the fact that ∞ + ∞ = ∞, and the fact that ∞ * ∞ = ∞) has nothing to do with the fact that ∞ is infinite, and everything to do with the fact that ∞ is a gap.
As one can probably guess from the name, gaps are generally between things. ∞ itself is the gap between positive finite and positive infinite numbers. More generally, anything with an absolute value smaller than it is positive, anything with a larger absolute value is infinite. The only things equal absolute values are ∞ and −∞, which are gaps rather than numbers.
It's kind of obvious, once you have that understanding, how gaps work. The thing is, I didn't. I thought that how gaps worked was how infinity worked.
Things would have been clearer if I'd spent more time thinking about 1/∞
But 1/∞ is zero.
Well . . . ish.
That's what I was taught and that's what I thought, but it's only true under certain circumstances. The key point is that zero is undirected.
What?
+0 = −0
So?
So if 1/∞ = 0 then:
1/∞ = 0 = −0 = −1/∞
which would mean that ∞ = −∞.
That's not a bad result. It actually works for various theories of stuff and it'll remove all those pesky discontinuities from things like the graph of the tangent function or y = 1/x (but not y = 1/x
^{2}), but usually we do not intend for ∞ to be equal to −∞, and so, in those cases, we have to accept that 1/∞ does not equal zero.
Then the question becomes what it does equal.
What we know about reciprocals helps here.
We're dealing with positive infinity, so we'll think in terms of positive numbers. We know that for x > y > 0 it is true that 1/y > 1/x > 0.
Since ∞ is greater than all positive numbers, 1/∞ must be less than them while still being greater than zero.
I've seen people call this result η (eta.)
The reason that thinking about this would have helped me is that η has many of the same properties as ∞ (η + η = η and η * η = η, for example) but it is most definitely not infinite. It's actually really, really small.
It is, in fact, the gap between the positive infinitesimals (numbers so small that you need infinite quantities of them just to add up to the smallest positive real numbers) and the rest of the positive numbers.
So very much not infinite, but the same as ∞ in many ways. Why? Because they're both gaps. So let's talk about properties of gaps.
The first thing about gaps is that they don't play well with others. When it comes to addition, they devour anything smaller (in absolute value) than them. That's why ∞ plus or minus any real number is still ∞.
This result is
not true of infinite things in general.
If we take the "simplest" infinite number, ω, and add one to it we get a new number which is larger than it is. How much larger? One larger. Just like we'd expect when adding one to any number.
It
is true of all gaps though. If we adjust for scale we see that the same rule that makes ∞ eat any finite number added to or subtracted from it makes η eat any infinitesimal added to or subtracted from it.
The second thing for gaps is that they play strangely with themselves.
If you add gaps of different scales the larger one eats the smaller one and you're left with just the larger one, but interesting things happen when you add gaps of the same scale.
A gap minus itself is zero, just like you'd expect. A gap plus itself is . . . the same gap. From that we can extend to the fact that a gap times any finite number is the same gap.
Going back to the same reasoning we use to prove that a gap plus itself is the same gap, we can also show that a gap times itself is the same gap.
Note that I'm not proving (or even demonstrating) any of this now. Just telling you about is. Regardless, so far we've got some decent properties.
For a gap "g" and any non-zero real number "x":
g − g = 0
g + g = g
g * x = g
g * g = g
This leads to some of the more extreme wonkyness you'll meet in ∞. In fact, you just need the first two to get things to a really weird place.
For example, I give you the infinite sheep theft and con:
Someone has done great evil to you and therefore, by way of revenge, you decide to steal all of their infinite (as in {1, 2, 3, . . .}) sheep and get away with it.
Here's how you do it.
When they're not looking you steal some of their sheep. You line them up and take every second one. You leave sheep number one, take sheep number two, leave sheep number three, and so forth.
How many sheep do you have?
Well your sheep were {2, 4, 6, ...} but that's not how one counts. Sheep 2 is clearly your first sheep, sheep four is clearly your second sheep, and it doesn't take too much thought to realize that sheep 2n is your nth sheep. So even though they are sheep {2, 4, 6, ...} you count them as sheep {1, 2, 3, ...} which happens to be infinity. The exact same infinity as the total number of sheep in their flock before you stole any.
You just took infinite sheep from an infinite flock, and both those infinities were the same infinity. Surely someone will notice!
Actually, no.
When your enemy counts up their sheep they're going to find that they have the same amount as before. We don't really care what order they count them in, it only matters that when one is counted it's taken out of the equation and not counted again, so we can put the sheep your enemy counts into the same order as when you put the sheep in a line.
Or we can just assume that the sheep always line up in the same order, moving forward to fill in any empty spaces resulting from removed sheep. Thus sheep one is right in front again, while sheep three has moved forward to stand behind, and so on.
Either way.
Your enemy has sheep {1, 3, 5, ...} but they don't know you took the even ones so when they count sheep three they'll be counting it as though it were sheep two, when they count sheep five they'll count it as though it is sheep three, and so forth. Sheep 2n-1 is counted as though it is sheep n.
At the end of the count they find they have {1, 2, 3, . . .} sheep, which is exactly how many they had before. (This in spite of the fact that their remaining sheep are really only the sheep that had odd numbers when you divided them.) They conclude none have been stolen.
You're in the clear.
But you wanted to take all their sheep as your revenge.
Thus begins the con.
You contrive a situation where your flock and their flock have to be combined for a time. Maybe they're put in quarantine in case they have the much feared sheep blight, maybe . . . anything really.
Whatever the case, your enemy and you don't exactly get along so to make sure you both get back the same number of sheep as you put in, you make a tally. You do it in public and above board and squeaky clean.
First they put in one of their sheep and make a tally mark (perhaps in their family crest or some such) then you put in one of your sheep and make a tally mark (perhaps in your initials or some such.) You repeat this until all the sheep are put in.
Some interesting things have happened here. One is that they have all the odd tally marks and you have all the even. The other is that sheep {1, 2, 3, 4, 5, 6, . . .} are the correct numbers again. The odds are the ones you didn't steal, the evens are the ones you did.
You make sure that you take your sheep first, perhaps complaining about how long the alternating thing took. (It took infinite time, by the way.)
Every second tally corresponds to one of your sheep. The deal was that you get the same number of sheep back, not the same sheep, so to simplify things you just grab the sheep from the front of the line.
At tally mark two, which is marked in such a way to let everyone know it really is yours, you take sheep 1, at tally mark 4 you take sheep 2, and so on with taking sheep n for mark 2n until you run out of tally marks.
You then go back home with your infinite sheep and have them graze at the infinite pasture.
When it comes time for your enemy to take their half of the sheep, they find there are none left. What the fuck?
Everyone knows that you didn't take extra sheep, you did the whole thing in public and you only took one sheep for each of your tally marks.
You're unimpeachable. They're out of sheep.
What happened? Well for mark two you took one of their sheep, for mark four you took on of yours, and you alternated until you ran out of marks. We've already shown that there are as many even numbers (the marks you used) as there are odd numbers (their sheep) and even numbers (your sheep) combined (the sheep you took) so you took all the sheep.
All the sheep?
ALL THE SHEEP!
In more formal language, you abused the fact that addition of gaps is non-associative to unbalance the equation at every stage of the game.
Recall that for any gap "g"
- g − g =0
- g + g = g
Short version:
For the theft you did (1), for the con you did (2):
(1) ∞ + (∞ − ∞) = ∞ + 0 = ∞
(2) (∞ + ∞) − ∞ = ∞ − ∞ = 0
(Short version is over.)
You were only able to reach those points (which are themselves perfectly sound mathematical statements) by ignoring the fact that, when discussing gaps the rules for addition are different than they are for numbers.
The associative property is the one that states:
a + (b + c) = (a + b) + c
That clearly doesn't work for gaps. (1) above is the same as the left side and it evaluated to ∞, (2) above is the same as the right side and it evaluated to 0. Since 0 ≠ ∞, we know that the associative property doesn't hold for addition of gaps.
(Long version of the explanation)
Again, for any gap "g"
- g − g =0
- g + g = g
In the beginning there were ∞ sheep.
Using property two
∞ = ∞ + ∞
which you achieved by separating the sheep into even and odd numbered sheep.
Then you took only the even numbered sheep
∞ + (∞ − ∞) = ∞ + 0 = ∞
which left them with ∞ sheep, the same amount they started with.
That was how you stole infinite sheep without anyone noticing.
Then you each created tally marks equivalent to your each of your sheep
∞ + ∞ = ∞ + ∞
Then when it came time to take the sheep back out, you combined the sheep (by taking them from the front rather than every other) and removed an amount equivalent to your tally marks from
that.
(∞ + ∞) − ∞ = ∞ − ∞ = 0
Thus leaving zero sheep.
⁂
Presumably everyone has stopped reading by now, so lets take things even more into the weeds. Not formal proofs, per se, but let's demonstrate things.
I've said that gaps eat everything smaller then themselves (under addition.) How do we know this?
Well, this is really easy to see with ∞.
Everything (positive) on one side is finite, everything on the other side is infinite, and we know certain things about such numbers.
For example: any finite number, any finite number, is a finite number.
For any finite numbers x and y,
if x + y = z
then z is a finite number.
This is where I ask you to imagine ∞ − y. Why minus instead of plus? Why "y" instead of "x"? Because I wrote a lot of this before thinking through simple ways to do things.
Now then, if ∞ − y is less than ∞, then it must be finite. (I should probably be using absolute values, but we aren't seriously expecting subtracting a finite number from will bring it to negative ∞ or beyond, are we?)
So take the finite number "∞ − y" and call it "x". Take the finite number "y" and call it "y". Plug it into the above note on finite numbers and we get x + y = z where z is a finite number.
Now solve for z.
Well that becomes (∞ − y) + y = z.
The associtive property might not work for gaps, but it's still as stellar as ever on numbers like y, so we can swap that to ∞ + (− y + y) = z, which is ∞ + 0 = z, and thus ∞ = z.
But z is finite and ∞ is infinite. Thus there's a contradiction. Thus ∞ - y can't be less than ∞ for any finite number y.
In symbols:
If ∞ − y < ∞,
then ∞ − y is finite.
We know that for x, y finite, and x + y = z
z is finite.
We set x = (∞ − y), y = y, and solve for z:
∞ − y + y = ∞ + (y − y) = ∞ + 0 = ∞
z = ∞
But z is finite and ∞ is infinite.
Contradiction
🡒🡐
Thus: ∞ − y is not less than ∞
(Or: thus ∞ − y ≥ ∞)
And, for the record, I'm somewhat pissed off that I can find this symbol "🡘" but not the inverse where the two arrow heads are colliding in the middle.
We're actually already done, we just have to say it.
Specifically:
Take the resulting expression
∞ − y ≥ ∞
and add y to both sides
∞ ≥ ∞ + y
For every finite number "y" there is a finite number "z" such that y = −z.
Substitute "−z" for "y"
∞ ≥ ∞ − z
We have already proven
∞ > ∞ − z is false
for all finite numbers z.
Therefore:
∞ = ∞ − z
Putting y back in to make things all neat and additive again:
∞ = ∞ + y, for all finite numbers y
Woo! Headache.
I could not find damn "not >" and "not <" signs. This is also the point where I realized I'd already used "z". Whatever.
The above shows what happens when we add things smaller than ∞ to ∞. We can, however, add things larger. ω + ∞ is a thing. It's the gap that separates ω + x from ω + y for all positive finite x and positive infinite y.
When not using infinite numbers or larger gaps, though, ∞ devours everything we might meaningfully throw at it. With two exceptions. Those exceptions are ∞ itself and -∞.
⁂
We made use of how ∞ + ∞ is equal to ∞ above. We can work it out here using the properties of gaps.
Say that ∞ + ∞ = [?]. Now pick any number between ∞ and [?]. Call that "x" (because we pretty much always call it "x" unless we have a good reason not to.)
x is larger than ∞ and so infinite. x is lower than [?]. So half of x must be smaller than ∞. Anything smaller than ∞ is finite. With x infinite and half of x finite, the question becomes, "What's the finite fucking number that becomes infinite when you double it?" Or we could say, "Solve for x/2."
Either way, it doesn't work. There can't be anything between ∞ + ∞ and ∞, which means they're the same thing. (Informal proof by contradiction for the win.)
We can do more or less the same thing for η, except upside down and backwards.
And we can do the same thing for multiplication. You can't have an infinite number with a finite square root.
⁂
Or, we could finally talk about eta a bit
Let η * η = [?]. We're dealing with positive things less than one, so if [?] doesn't equal η then it must be smaller. (0 ≤ |x*x| ≤ |x| for all x such that |x| ≤ 1) With that in mind, choose a y such that [?] ≤ y ≤ η.
Consider √y. Since [?] ≤ y, η ≤ √y.
As I noted, this turns out to be the same thing we did with addition. Everything larger than η is not infinitesimal, everything smaller (or equal) is. You can't have a non-infinitesimal whose square is infinitesimal. Thus √y = η = y = [?]
The key point is that since [?] = η * η we've just shown η * η = η.
Not that that really matters to us.
Addition is where it's at.
Or at least it was back when the sheep theft came right after this.
⁂
Beyond stealing sheep it's worth noting that since g + g = g we can collapse any finite length of g-only addition.
So g + g + g + g + . . . + g = g. Thus g*n = g, for a any natural number n. We can actually expand that to g*x = g provided that x is comparable to 1. Comparable, here, means that if the absolute value of x is less than one there's some natural number (we use those a lot) for which |n*x| > 1, and if the absolute value of x is greater than one there's a natural number n such that n*1 > x. If I'd used "y" instead of "1" then I'd have to stipulate that the inequalities used "|y|" instead of "y".
The infinitesimals are not comparable to the reals which are not comparable the infinites.
⁂
Anyway, back on point, most of the weird properties I associated with infinity came from attributes of gaps, such as that. Not all, though. The thing about parallel lines converging, for example, is totally unrelated to gaps.
Why does any of this matter?
It turns out that there's a wonderful and rich number system that allows you to go beyond infinity (
way beyond), as well as dive into the infinitesimals. There's just one problem with it: it's riddled with gaps.
To start off, consider η and ∞ once again. η*x is the gap below everything comparable to x, ∞*x is the gap above everything comparable to x. Since η and ∞ are themselves times one, they're the gaps below and above (respectively) everything comparable to the positive reals. You can also multiply them by any power of ω. That'll get you uncountably infinite gaps right there.
There is a question of "Where does this end?" as the η*ω
^{x}s get smaller and smaller and the ∞*ω
^{y}s get larger and larger. (We could ask for the reverse, but it makes intuitive sense to ask as the lower bounds get to be of progressively smaller sets and the upper bounds become of progressively larger ones.) Well the η*ω
^{x}s approach the gap between zero and all of the positive numbers. The ∞*ω
^{y}s approach a thing called "On". It is the gap above all numbers (no matter how infinitely infinite their infinitude) and the reciprocal of the little gap.
The little gap is more meaningful to us when talking about how many gaps there are. Since the gap between zero and the positives is less than the absolute value of x for all non-zero x, it can be added to or subtracted from every single number other than zero to create a new gap, with itself and negative itself basically standing in for adding it to zero, the only thing it can eat. In other words, for every surreal number there are two of these gaps.
Those are in addition to the already infinite ones we were discussing by multiplying η or ∞ by powers of ω. And, about those, every one of them, indeed every pure gap other than On and -On, is smaller in absolute value than infinitely many numbers. The infinite η / ∞ times powers of ω gaps plus or minus any legal number (of which there will be infinite for whichever η or ∞ gap you're talking about) will produce brand new gaps. At this point I shouldn't have to point out there will be infinite of them.
It turns out that we don't have a word for a collection so damn large it can contain all of the gaps. This even though we do have words for infinite sets and proper classes, which are larger than infinite sets.
This seems to be where my rambling ran out.
⁂
⁂ ⁂
* We talked about parallel lines converging at infinity (sometimes it was a point, sometimes it was a line).
We noted how there are different sizes of infinity by looking at how there are more irrational numbers than rational ones even though there's a rational number between every two irrationals (and vice versa) so it would seem impossible for there to be more of one than the other.
One professor shared with us the story of the infinite hotel** which, though full, could accommodate (countably) infinite additional guests by shuffling rooms.
We demonstrated that there were the same number of even numbers as there were odd plus even numbers combined.
We showed that there were the same number of rational numbers (q/p for integers q and p) as there were natural numbers (1, 2, 3, . . .) in spite of the fact that natural numbers make up a tiny part of the rational numbers (the bottom of the fraction is always jammed at 1.)
We showed that there were infinitely more irrational numbers than there were rational ones, this even though we also showed that between any two irrational numbers there was a rational one. (
You try making a sequence of things from sets X and Y such that there's an element of Y between every two elements of X and have the number of Y things used be
infinitely smaller than the number of X things used.)
We did all kinds of weird shit.
One definition of infinity is that a set has infinite elements when it's possible for a proper subset of it to have a one to one correspondence with the whole set.
Proper subset just means that it's not the whole set itself. So if we've got the set {cow, cheese} the subsets are {cow}, {cheese}, {cow, cheese} and the empty set {}, The
proper subsets are {cow}, {cheese}, and the empty set {}, because we get all of them by removing actual stuff from the original set.
One to one correspondance is why tally marks and counting on your fingers both work. For every element of one set there's one and only one element of the other.
For each tally mark there is a sheep, for example, and as of the most recent rewrite I already did the sheep thing above.
The sheep thing, as demonstrated above, shows that:
infinity minus infinity is infinity
and
infinity minus infinity is zero
Like I said, the point at which things stop making sense.
⁂
** It was not as formal, or German, as
Hilbert's original, and went something like this:
The hotel has rooms 1, 2, 3, and so on to infinity. (We call this "countable infinity, by the way.) They're all full. In spite of this, the sign reads, "Vacancy".
A new guest arrives, and one of the people at the counter is all, "I told you we should have lit the 'No' part of the sign,"
The other just says, "No problem. We told our guests they might be forced to move from one room to the next. We'll put the new person in room one, have the person in room one move to room two, the person in room two move to room three . . . "
And the new guest is thus added to the roster of guests. The same can happen if a million, a billion, a googolplex, or whatever show up. Any finite number is fine, just move all the existing guests to the room [that number]+1.
Then infinite new guests arrive, first person is again negative, second person again says, "No problem."
"We can't move the guest in room one to room infinity plus one. Infinity plus one is infinity!"
"No, no. Here's what we do. We move the guest in room one to room two, then--"
"Where does the guest in room two go?"
"To room four. And the guest in room three goes to room six."
"We move guest in room N to room 2N."
"Exactly. Then we put the first guest new guest in room 1, the second in room 3, and--"
"New guest N to room 2N-1."
"You got it."
And so the infinite hotel, which was already full, gives empty rooms to infinite new guests without leaving any of the old guests without a room of their own.
Then there are buses with (countably) infinite guests on them, and how many buses? 1, 2, 3, so on, infinity!
Still not a problem.
Then the irrationals arrive and the people at the desk say, "Fuck it."