Infinities are one of the most complex and confounding topics in mathematics and they lead to an absurd number of paradoxes. However, many of the paradoxes turn out to be variations on the same theme once you dig into what is actually happening. I will provide informal hints on how surreal numbers could help us solve some of these paradoxes, although the focus on this post is primarily categorisation, so please don't mistake these for formal proofs. I'm also aware that simply noting that a formalisation provides a satisfactory solution doesn't philosophically justify its use, but this is also not the focus of this post. I plan to update this post to include new infinite paradoxes that I learn about, including ones that are posted in the comments.
Infinitarian Paralysis: Suppose there are an infinite number of people and they are happy so there is infinite utility. A man punches 100 people and destroys 1000 utility. He then argues that he hasn't done anything wrong as there was an infinite amount utility before and that there is still an infinite utility after. What is wrong with this argument?
If we use cardinal numbers, then we can't make such a distinction. However, if we use surreal numbers, then n+1 is different from n every when n is infinite.
Paradox of the Gods: "A man walks a mile from a point α. But there is an infinity of gods each of whom, unknown to the others, intends to obstruct him. One of them will raise a barrier to stop his further advance if he reaches the half-mile point, a second if he reaches the quarter-mile point, a third if he goes one-eighth of a mile, and so on ad infinitum. So he cannot even get started, because however short a distance he travels he will already have been stopped by a barrier. But in that case no barrier will rise, so that there is nothing to stop him setting off. He has been forced to stay where he is by the mere unfulfilled intentions of the gods"
Let n be a surreal number representing the number of Gods. The man can move: 1/2^n distance before he is stopped.
Two Envelopes Paradox: Suppose that there are two sealed envelopes with one having twice as much money in it as the other such that you can't see how much is in either. Having picked one, should you switch?
If your current envelope has x, then there is a 50% chance of receiving 2x and a 50% chance of receiving x/2, which then creates an expected value of 5x/4. But then according to this logic we should also want to switch again, but then we'd get back to our original envelope.
This paradox is understood to be due to treating a conditional probability as an unconditional probability. The expected value calculation should be 1/2[EV(B|A>B) + EV(B|A<B)] = 1/2[EV(A/2|A>B) + EV(2A|A<B)] = 1/4 EV(A|A>B) + EV(A|A<B)
However, if we have a uniform distribution over the positive reals then arguably EV(A|A>B) = EV(A|A<B) in which case the paradox reoccurs. Such a prior is typically called an improper prior as it has an infinite integral so it can't be normalised. However, even if we can't practically work with such a prior, these priors are still relevant as it can produce a useful posterior prior. For this reason, it seems worthwhile understanding how this paradox is produced in this case
Suppose we uniformly consider all numbers between 1/n and n for a surreal n and we double this to generate a second number between 2/n and 2n. We can then flip a coin to determine the order. This now creates a situation where EV(A|A>B) = n+1/n and EV(A|A<B) = (n+1/n)/2. So these expected values are unequal and we avoid the paradox.
Sphere of Suffering: Suppose we have an infinite universe with people in a 3-dimensional grid. In world 1, everyone is initially suffering, except for a person at the origin and by time t the happiness spreads out to people within a distance of t from the origin. In world 2, everyone is initially happy, except for a person at the origin who is suffering and this spreads out. Which world is better? In the first world, no matter how much time passes, more people will be suffering than happy, but everyone becomes happy after some finite time and remains that way for the rest of time.
With surreal numbers, we can use l to represent how far the grid extents in any particular direction and q to represent how many time steps there are. We should then be able to figure out whether q is long enough that the majority of individual time slices are happy or suffering.
Suppose we have an infinite set of finite numbers (for example the natural numbers or the reals), for any finite number x, there exists another finite number 2x. Non-formally, we can say that each finite number is in the first half; indeed the first 1/x for any integer x. Similarly, for any number x, there's always another number x+n for arbitrary n.
The canonical example is the Hilbert Hotel. If we have an infinite number of rooms labelled 1, 2, 3... all of which are full, we fit an extra person in by shifting everyone up one room number. Similarly, we can fit an infinite number of additional people in the hotel by sending each person in room x to room 2x to free up all of the odd rooms. This kind of trick only works because any finite-indexed person isn't in the last n rooms or in the last-half of rooms.
If we model the sizes of sets with cardinal numbers, then all countable sets have the same size. In contrast, if we were to say that there were n rooms where n is a surreal number, then n would be the last room and anyone in a room > n/2 would be in the last half of rooms. So by modelling these situations in this manner prevents last half paradoxes, but arguably insisting on the existence of rooms with infinite indexes changes the problem. After all, in the original Hilbert Hotel, all rooms have a finite natural number, but the total number of rooms are infinite.
Galileo's Paradox: Similar to the Hilbert Hotel, but notes that every natural number has a square, but the that only some are squares, so the number of squares must be both at least the number of integers and less than the number of integers.
Bacon's Puzzle: Suppose that there is an infinitely many people in a line each with either a black hat or a white hat. Each person can see the hats of everyone in front of them, but they can't see their own hat. Suppose that everyone was able to get together beforehand to co-ordinate on a strategy. How would they be able to ensure that only a finite number of people would guess the colour of their hat incorrectly? This possibility seems paradoxical as each person should only have a 50% chance of guessing their hat correctly.
We can divide each infinite sequence into equivalence classes where two sequences are equivalent if they stop differing after a finite number of places. Let's suppose that we choose one representative from each equivalence class (this requires the axiom of choice). If each person guesses their hat as per this representative, after a finite number of places, each person's guess will be correct.
This paradox is a result of the representative never last differing from the actually chosen sequence in the last half of the line due to this index having to be finite.
Trumped: Donald Trump is repeatedly offered two days in heaven for one day in hell. Since heaven is as good as hell is bad, Trump decides that this is a good deal and accepts. After completing his first day in hell, God comes back and offers him the deal again. Each day he accepts, but the result is that Trump is never let into heaven.
Let n be a surreal number representing how many future time steps there are. Trump should accept the deal as long as the day number is less than n/3
St Petersberg Paradox: A coin is tossed until it comes up heads and if it comes up heads on the nth toss, then you win 2^n dollars and the game ends. What is the fair price for playing this game? The expected value is infinite, but if you pay infinity, then it is impossible for you to win money.
If we model this with surreals, then simply stating that there is potentially an infinite number of tosses is undefined. Let's suppose first that there is a surreal number n that bounds the maximum number of tosses. Then your expected value is $n and if we were to pay $n/2, then we'd expect to make $n/2 profit. Now, setting a limit on the number of tosses, even an infinite limit is probably against the spirit of the problem, but the point remains, that paying infinity to play this game isn't unreasonable as it first sounds and the game effectively reduces to Pascal's Wager.
Trouble in St. Petersberg: Suppose we have a coin and we toss it until we receive a tails and then stop. We are offered the following deals:
- Offer 1: Lose $1 if it never lands on tails, gain $3 if it lands on tails on toss 1
- Offer 2: Lose $4 if if lands on tails on toss 1, gain $9 if it lands on tails on toss 2
- Offer 2: Lose $10 if if lands on tails on toss 2, gain $13 if it lands on tails on toss 3
And so on, when we calculate the losses by doubling and adding two and the gains by doubling and adding 3.
Each deal has a positive expected value, so we should accept all the deals, but then we expect to lose in each possibility.
In the finite case, there is a large amount of gain in the final offer and this isn't countered by a loss in a subsequence bet, which is how the EV ends up being positive despite being negative in every other case. The paradox occurs for infinities because there is no last n with an unmitigated gain, but if we say that we are offered n deals where n is a surreal number, then deal n will have this property.
Dice Room Murders: Suppose that a serial killer takes a man hostage. They then roll a ten-sided dice and release them if it comes up 10, killing them otherwise. If they were killed, then the serial killer repeats taking twice as many hostages each time until they are released. It seems that each person taken hostage should have a 1/10 chance of surviving, but there are always more people who survive than die.
There's an infinitesimal chance that the dice never comes up a 10. Including this possibility, approximately 50% of people should survive, excluding it the probability is 1/10.
This is a result of a biased sample. To see a finite version, imagine that there are two people. One tosses a coin. If it's not heads, the other tosses a coin too. The person-flip instances will be 50/50 heads/tails, but if you average over people before averaging over possibilities, you'll expect most of the flips to be heads. How you interpret this depends on your perspective on SSA and SIA. In any case, nothing to do with infinities.
Ross-Littlewood Paradox: Suppose we have a line and at each time period we add 10 balls to the end and then remove the first ball remaining. How many balls remain at the end if we perform this an infinite number of times?
Our first intuition would be to answer infinite as 9 balls are added in each time period. On the other hand, the nth ball added is removed at time 10n, so arguably all balls are removed. Clearly, we can see that this paradox is a result of all balls being in the first 1/10.
Soccer Teams: Suppose that there are two soccer teams. Suppose one team has players with abilities ...-3,-2,-1,0,1,2,3... Now lets suppose a second team started off equally as good, but all the players trained until they raised their ability levels by one. It seems that the second team should be better as each player has strictly improved, but it also seems like it should be equally good as both teams have the same distribution.
With surreal numbers, suppose that the teams originally have players between -n and n in ability. After the players have trained, their abilities end up being between -n+1 and n+1. So the distribution ends up being different after all.
Positive Soccer Teams: Imagine that the skill level of your soccer team is 1,2,3,4... You improve everyone's score by two which should improve the team, however your skill levels are now 3,4... Since your skill levels are now a subset of the original it could be argued that the original team is better since it has two addition (weakly positive) players assisting.
Can God Pick an Integer at Random? - Suppose that there are an infinite amount of planets labelled 1,2,3... and God has chosen exactly one. If you bet that God didn't pick that planet, you lose $2 if God actually chose it and you receive $1/2^n. On each planet, there is a 1/∞ chance of it being chosen and a (∞-1)/∞ chance of it not being chosen. So there is an infinitesimal expected loss and a finite expected gain, given a positive expected value. This suggests we should bet on each planet, but then we lose $2 on one planet win less than $1 from all the other planets.
Let surreal n be the number of planets. We expect to win on each finite valued planet, but for surreal-valued planets, our expected gain becomes not only infinitesimal, but smaller than our expected loss.
Banach-Tarski Paradox: This paradox involves describing a way in which a ball can be divided up into five sets that can then be reassembled into two identical balls. How does this make any sense? This might not appear like an infinite paradox as first, but this becomes apparent once you dig into the details.
Let's first explain how the proof of this paradox works. We divide up the sphere by using the free group of rank 2 to create equivalence classes of points. If you don't know group theory, you can simply think of this as combinations of where an element is not allowed to be next to its inverse, plus the special element 1. We can then think of this purely in terms of these sequences.
Let S represent all such sequences and S[a] represent a sequence starting with a. Then .
In other words, noting that there is a bijection between all sequences and all sequences starting with any symbol s, we can write
Breaking down similarly, we get:
In other words, almost precisely two copies of itself apart from 1.
However, instead of just considering the sequences of infinite length, it might be helpful to assign them a surreal length n. Then S[a] consists of "a" plus a string of n-1 characters. So S[aa] isn't actually congruent to S[a] as the former has n-2 addition characters and the later n-1. This time instead of the paradox relying on there being that no finite numbers are in the last half, it's relying on there being no finite length strings that can't have either "a" or "b" prepended in front of them which is practically equivalent if we think of these strings as representing numbers in quaternary.
The Headache: Imagine that people live for 80 years. In one world each person has a headache for the first month of their life and are happy the rest, in the other, each person is happy for the first month, but has a headache after that. Further assume that the population triples at the end of each month. Which world is better? In the first, people live the majority of their life headache free, but in the second, the majority of people at any time are headache free.
If we say that the world runs for t timesteps where t is a surreal number, then the people in the last timesteps don't get to live all of their lives, so it's better to choose the world where people only have a headache for the first month.
The Magic Dartboard: Imagine that we have a dartboard where each point is colored either black or white. It is possible to construct a dartboard where all but measure 0 of each vertical line is black and all but measure 0 of each horizontal line is white. This means that that we should expect any particular point to be black with probability 0 and white with probability 0, but it has to be some color.
One way of constructing this situation is to first start with a bijection f from [0, 1] onto the countable ordinals which is known to exist. We let the black points be those ones where f(x) < f(y). So given any f(y) there are only a countable number of ordinals less than it, so only a countable number of x that are black. This means that the measure of black points in that line must be 0, and by symmetry we can get the same result for any horizontal line.
We only know that there will be a countable number of black x for each horizontal line because f(x) will always be in the first 1/n of the ordinals for arbitrary n. If on the other hand we allowed f(x) to be say in the last half of countable ordinals, then for that x we would get the majority of points being black. This is distinct from the other paradoxes in this section as for this argument to be correct, this theorem would have to be wrong. If we were to bite this bullet, it would suggest any other proof using similar techniques might also be wrong. I haven't investigated enough to conclude whether this would be a reasonable thing to do, but it could have all kinds of consequences.
The canonical example is Thomson's Lamp. Suppose we have a lamp that is turned on at t=-1, off at t=-1/2, on at t=1/4, ect. At t=0, will the lamp be on or off?
With surreal numbers, this question will depend on whether the number of times that the switch is pressed is represented by an odd or even Omrific number, which will depend in a relatively complex manner on how we define the reals.
Grandi's Series: What is the sum of 1-1+1-1...?
Using surreal numbers, we can assign a length n to the series as merely saying that it is infinite lacks resolute. The sum then depends on whether n is even or odd.
This is one class of paradoxes that surreal numbers don't help with as surreals don't have a largest finite number or a smallest infnity.
Satan's Apple: Satan has cut a delicious apple into infinitely many pieces. Eve can take as many pieces as she likes, but if she takes infinitely many pieces she will be kicked out of paradise and this will outweigh the apple. For any finite number i, it seems like she should take that apple piece, but then she will end up taking infinitely many pieces.
I find this paradox the most troubling for attempts to formalise actual infinities. If we actually had infinitely many pieces, then we should be able to paint all finitely numbered pieces red and all infinitely numbered pieces blue, but any finite number plus one is finite and any infinite number minus one is infinite, so it doesn't seem like we can have a red and blue piece next to each other. But then, what does the boundary look like.
These "paradoxes" may point to interesting mathematical phenomenon, but are so easily resolved that they hardly deserve to be called paradoxes.
Gabriel's Horn: Consider rotating 1/x around the x-axis. This can be proven to have finite volume, but infinite surface area. So it can't contain enough paint to paint its surface.
It only can't paint its surface if we assume a fixed finite thickness of paint. As x approaches infinity the size of the cross-section of the horn approaches 0, so past a certain point, this would make the paint thicker than the horn.
Bertrand Paradox: Suppose we have an equilateral triangle inscribed in a circle. If we choose a chord at random, what is the probability that the length of chord is longer than a side of the triangle.
There are at least three different methods that give different results:
- Picking two random end points gives a probability of 1/3
- Picking a random radius then a random point on that radius gives probability of 1/2
- Picking a random point and using it as a midpoint gives 1/4
Now some of these method will count diameters multiple times, but even after these are excluded, we still obtain the same probabilities.
We need to bite the bullet here and say that all of these probabilities are valid. It's just that we can't just choose a "random chord" without specifying this more precisely. In other words, there isn't just a single set of chords, but multiple that can be defined in different ways.
Zeno's Paradoxes: There are technically multiple paradoxes, but let's go with the Dichtomy Paradox. Before you can walk a distance, you must go half way. But before you can get halfway, you must get a quarter-way and before that an eight of the way. So moving a finite distance requires an infinite number of tasks to be complete which is impossible.
It's generally consider uncontroversial and boring these days that infinite sequences can converge. But more interesting, this paradox seems to be a result of claiming an infinite amount of time intervals to diverge, whilst allowing an infinite number of space intervals to converge, which is a major inconsistency.
Skolem's Paradox: Any countable axiomisation of set theory has a countable model according to the Löwenheim–Skolem theorem, but Cantor's Theorem proves that there must be an uncountable set. It seems like this confusion arises from mixing up whether we want to know if there exists a set that contains uncountably many elements or if the set contains uncountably many elements in the model (the corresponding definition of membership in the model only refers to elements in the model). So at a high level, there doesn't seem to be very much interesting here, but I haven't dug enough into the philosophical discussion to verify that it isn't actually relevant.