[Math Lair] Paradoxes

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A paradox is a statement that goes against our intuition but may be true, or a statement that is or appears to be self-contradictory. Several well-known logical, mathematical, and other paradoxes are listed below. Many mathematical paradoxes fall into one of two categories: either they result from the counter-intuitive properties of infinity, or are a result of self-reference. This page lists several well-known logical and mathematical paradoxes.

Well-Known Paradoxes

Skip to: Zeno's Paradoxes | Russell's Paradox | Grelling's Paradox | Barber Paradox | Cretan Liar Paradox | Thomson Lamp Paradox | G. G. Berry's Paradox | Galileo's Paradox | Hilbert's Hotel | Rearrangement Paradoxes | Shopkeeper's Paradox | Newcomb's Paradox | Sorites Paradox | St. Petersburg Paradox | Raven Paradox | Banach-Tarski Paradox | Grue-Bleen Paradox | Birthday Paradox | Two Envelopes Paradox | Denotation Paradox | Paradox of the Inclusive Map | Hintikka's Paradox.

Zeno's Paradoxes
Is motion possible? Most of us seem to think so, but Zeno gives several arguments why it isn't, as motion seems to require an infinite number of steps.
Russell's Paradox
Take the set of all sets that are not members of themselves. Is it a member of itself? If it is, it isn't. If it isn't, it is. This was a significant paradox in set theory in the early 1900s. See also vicious circle paradoxes.
Grelling's Paradox
A version of Russell's Paradox using words. Some adjectives are self-descriptive, like "tiny", "unhyphenated", and "pentasyllabic". On the other hand, other adjectives are not self-descriptive, like "monosyllabic", "big", "tasty", and "incomplete". Call the self-descriptive adjectives autological, and the non-self-descriptive adjectives heterological. Now, is "heterological" autological or heterological? If it is, then it isn't. If it isn't, then it is. Either way, there's a paradox. I've occasionally seen it spelled "Greeling's Paradox," but the originator's name was Kurt Grelling.
Barber Paradox
In the town barber shop, the (male) barber puts a sign up which states that he shaves all men in the town who don't shave themselves, and only those men. Does the barber shave himself or not? If he does, he doesn't. If he doesn't, he does. This paradox is an easy-to-understand illlustration of Russell's paradox.
Barber Shop Paradox
The Barber Shop paradox or the Barbershop paradox can refer either to the Barber paradox above, or the Shopkeeper's paradox below.
Cretan Liar Paradox
Similar to Russell's paradox, this paradox dates back to the ancient Greeks. One variant of this is as follows: Someone states: "This statement is false". Is the statement false or not? If it's false, then the statement is true. If it's true, it's false.
Thomson Lamp Paradox
We have a perfect machine for turning a light on and off. First, we have the light on for one minute, after which it is turned off for one half of a minute. Then it is on again for one fourth of a minute and off for one eighth of a minute. This continues with the light turned on or off after one half of the preceding time period. After two full minutes an infinite sequence of offs and ons will have occurred. At this time, will the light be on or off? See supertasks for further discussion of this paradox.
G.G. Berry's Paradox
We can classify the integers based on the smallest number of syllables in English necessary to describe them. For example, 6 could be described in one syllable as "six," while 729 could be described in two syllables as "nine cubed." Since there are only a finite number of words in the English language, only a finite (although possibly very large) number of numbers can be described with a given number of syllables. Specifically, for the purpose of this argument, only a finite number of positive integers can be described using 18 or fewer syllables. Therefore, there must be a least integer not describable using less than 19 syllables. Now, consider the set of all positive integers that require at least nineteen syllables to describe them. This set will have a smallest element. However, we can describe this integer as the "least integer not describable using less than nineteen syllables", which is a description of eighteen syllables! Therefore, there is no least integer that requires nineteen syllables to be described, which results in a contradiction.
Galileo's Paradox
There are as many square numbers as there are integers and vice versa. This is exhibited in the correspondence
1 <—> 1
2 <—> 4
3 <—> 9
4 <—> 16
5 <—> 25
   . . .
But how is this possible when not every number is square? The answer is that both sets are infinite sets with the same cardinality.
Hilbert's Hotel
Imagine a hotel with an infinite number of rooms, all of which are full. An infinite number of guests arrive and, although the hotel is fully booked, it is possible to make room for all of the infinite number of guests.
Rearrangement Paradoxes
These paradoxes assert that the sum of an infinite series may be changed by rearranging its terms. For example, 0 = (1 - 1) + (1 - 1) + (1 - 1) + . . . = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + . . . = 1 + 0 + 0 + 0 + . . . = 1.0 = 1.
The fallacy is that infinite series that are not convergent do not have a sum. See mathematical fallacies for similar fallacies.
Shopkeeper's Paradox
Allen owns a barber shop. There are two employees, Brown and Carr. At least one of the three must be in the shop at all times (rule #1). Allen was ill recently, and is nervous about going out alone, so he always takes Brown with him (rule #2). Now, start by assuming that Carr is out. Therefore, by rule #1, if Allen is out then Brown must be in. However, by rule #2, if Allen is out then Brown must be out. Because our assumption that Carr is out leads to a contradiction, that must mean that our assumption is false and Carr can never leave the store. This doesn't make any sense, though, because it seems perfectly reasonable that Carr could leave the store without violating either rule #1 or rule #2. The resolution is that the definition of "if... then" used in propositional logic is different from that used in everyday language, and the statements "if Allen is out then Brown must be in" and "if Allen is out then Brown must be out" are not in fact contradictory. This paradox was originally formulated by Lewis Carroll. Read the original formulation of the paradox.

This unintuitive definition of "if... then" can be used to create many confusing statements. Consider the statement "If I'm not mistaken, then UFOs exist." Now, by the rules of logic, this statement must be true! By the definition of "if... then", the only way the statement could be false is if you're not mistaken, and UFOs don't exist. But this is a contradiction, since if you're not mistaken, then UFOs exist. The resolution of this unintuitive state of affairs is that the truth value of the sentence as a whole tells us nothing about whether UFOs in fact exist, just as the above rules tell us nothing about who is actually inside the store.

Newcomb's Paradox
Two closed boxes, B1 and B2, are on a table. B1 contains $1,000. You don't know what B2 contains. You may choose either to (a) take the contents of both boxes, or (b) take only what is in B2. Some time beforehand, an entity who can make highly accurate predictions about human behaviour (and whom you know is able to do so) has made a prediction as to your choice. If the entity expects you to choose both boxes, he has left B2 empty. If he expects you to take only B2, he has put $1 million in it. So, should you choose (a) or (b)? The paradox lies in the fact that there are compelling reasons for choosing either. On the one hand, it seems highly likely that you will get the $1,000,000 if you select B2, and only $1,000 if you select both boxes. However, either the money is already in B2 or it isn't. Taking both boxes will not cause the $1 million to disappear from B2 if it is already there. So, regardless of whether the money is in B2 or not, you will make $1,000 more by selecting both boxes.
Sorites Paradox
The name of this paradox comes from the Greek word for "heap." Consider the two fairly intuitive statements:
  1. A single grain of sand is not a heap.
  2. Adding a single grain of sand to something that is not a heap of sand won't turn it into a heap.
Now, by the first statement, a single grain of sand is not a heap. Now, if we add a single grain of sand to that, this won't be a heap either. So, neither will three grains of sand, or four, and so on and so on. If we carried this far enough, we would then have to say that a million grains of sand is not a heap, or a billion, or more, which doesn't make sense.

This paradox could be expressed in other ways. Consider the following:

  1. 1 is not a large number.
  2. Adding 1 to a number that is not large does not make it large.
So, by this reasoning, 2 would not be a large number, nor 3, nor 4, but as you go on you would find that 1,000,000 is not a big number, nor is 1,000,000,000, and so on. This paradox can be resolved by using a fuzzy concept of what "large numbers" or "heaps" are.
St. Petersburg Paradox
Flip a coin until it comes up heads. If it comes up heads on the first toss, you win $1. If it first comes up heads on the second toss, you win $2; if heads first appears on the third toss, you win $4, on the fourth toss you win $8 and, in general, if heads first appears on the nth toss, you win 2n − 1. What is your expectation for this game? Doing the math, the expectation is infinite, which doesn't make much sense.
Raven Paradox
This paradox, also known as the "paradox of confirmation", is a paradox in inductive logic. Take the hypothesis "All ravens are black". One could find evidence to support that hypothesis by finding one or more black ravens. However, by the rules of logic, the hypothesis "All ravens are black" is equivalent to "All non-black objects are non-ravens." So, a red apple or a blue jacket or a green leaf would support the statement "All non-black objects are non-ravens", and so would support the equivalent statement "All ravens are black." Furthermore, these objects would also support the hypothesis "All ravens are white," so the same observation can support two contradictory hypotheses.
Banach-Tarski Paradox
A sphere can be cut into five "pieces" (actually, non-measurable sets of points), which can then be reassembled into two spheres, each with volume equal to the original sphere. The paradox illustrates possible issues with the Axiom of Choice, a controversial axiom in Zermelo-Fraenkel set theory. One way of resolving the paradox would be to note that the "pieces" are highly abstract sets that, in the real world, would require infinitely precise cuts, so that the process could not be performed on any physical object. Looking at the spheres as a collection of points and not as physical objects helps to resolve the paradox, as the two spheres have the same number of points, specifically an uncountably infinite number, as one sphere does.
Grue-Bleen Paradox
Call an object grue if it has the property of being green at every moment before January 1, 2100, and blue afterwards. Similarly, an object is bleen if it is blue before January 1, 2100, and green afterwards. Take an emerald, for example. We observe its colour to be green. This would support both the statements "This emerald is green" and "This emerald is grue". But why are we more likely to claim that the emerald is green, and not grue? Well, using Occam's razor, it makes more sense to select the simpler hypothesis, that the emerald is green. But what if we imagine some tribe whose language doesn't have words for "green" and "blue", but only "grue" and "bleen"? To describe an object as "green", they would have to describe it as being "grue before January 1, 2100 and bleen afterward." Wouldn't they find describing the emerald as grue to be the simpler hypothesis? One way of resolving this paradox is that the tribal language is not equivalent to English or other languages; it is artificial. To teach young children what "green" and "blue" are, it is sufficient to simply show them a number of green and blue objects. How could the tribe possibly teach young children what "grue" and "bleen" are?
Birthday Paradox
How many people would you need to choose randomly before there is at least a 50% chance of two of them sharing a birthday? Surprisingly enough, the answer is only 23.
Two Envelopes Paradox
You are asked to select one of two envelopes, one of which contains twice as much money as the other. You are then asked whether you would like to switch. Since you could make twice as much money as you already have by switching, but only lose half as much money, it's to your advantage to switch... but now that you've exchanged envelopes, the same reasoning would lead you to switch again, and again, and again.
Denotation Paradox
One example of this paradox is as follows: 343 comprises three symbols. Since 343 = 7³, 7³ comprises three symbols. This paradox involves a confusion between a number (in this case 343) and its representation ("343").
Paradox of the Inclusive Map
In the inclusive map paradox, imagine that an area in England has been perfectly levelled off, and a map-maker traces a perfect map of England on the soil. The job is so perfect that there is no detail of England that cannot be found on the map. So, the map should contain a map of the map, which should contain a map of the map of the map, and so on, to infinity.
Hintikka's Paradox
It seems unusual to state that anything that is impossible is wrong, but Hintikka uses the rules of logic to apparently show this to be the case.

See also: Vicious Circle Paradoxes, Dissection Puzzles.

Sources used (see bibliography page for titles corresponding to numbers): 5, 56, 58.