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 and famous mathematical paradoxes, as well as some
non-mathematical paradoxes, are listed below. Note that 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.
Some Famous Paradoxes:
- Zeno's Paradox
- Is motion possible? Most of us seem to think so, but Zeno disagrees.
- 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.
- Greeling'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.
- Barber Paradox
- A version of Russell's 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?
- Barber Shop Paradox
- Also written "Barbershop paradox", this 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 it 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?
- G.G. Berry's Paradox
- We can classify the integers based on the smallest number of
syllables in English necessary to describe them. Consider the
set of all 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.
- 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 examples like this as well as other 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.
- Newcomb's Paradox
- Consider the following scenario: Two closed boxes, B1 and B2, are on
a table. B1 contains $1,000. B2 contains either nothing or $1 million
(you do not know which). You may choose either to (a) take the contents
of both boxes, or (b) take only what is in B2. Some time before the test,
an entity who is able to make highly accurate predictions about your
decisions has made a prediction about what you will decide. If the
entity expects you to choose both boxes, he has left box B2 empty. If he expects you to take only
B2, he has put $1 million in it (assume that, if the entity believes that you will somehow avoid making a choice,
such as by flipping a coin, he will leave B2 empty).
The paradox lies in the fact that
there are valid reasons for choosing either (a) or (b). If you
take both boxes, the entity will almost certainly have anticipated that
and left B2 empty, whereas if you take only B2, the entity will almost
certainly have anticipated that, and put $1 million in B2,
so you should take only B2. However, either the money is
already in B2 or it isn't. The contents of B2 will not change once you make your choice.
So, whether there is $1 million in B2 or not, you will always make
$1,000 more by choosing both boxes.
theory, we might say that there is a conflict between the
"expected-utility principle" and the "dominance principle". Assuming
that the being is able to predict with near certainty, the expected
utility of taking only box B2 is much larger (you're very likely to
take $1 million, whereas if you take both boxes, you're very likely
to only get $1000). On the other hand, no matter what is in the boxes,
the option to take both boxes dominates the option to only take box B2.
- Sorites Paradox
- The name of this paradox comes from the Greek word for "heap." Consider the two fairly intuitive statements:
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.
- A single grain of sand is not a heap.
- Adding a single grain of sand to something that is not a heap of sand won't turn it into a heap.
This paradox could be expressed in other ways. Consider the following:
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.
- 1 is not a large number.
- Adding 1 to a number that is not large does not make it large.
- St. Petersburg Paradox
- This paradox describes a game that apparently has an infinite expectation, but most people wouldn't pay anywhere near that to play.
- 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
- It is possible to cut a sphere 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. When we start looking at the spheres as a collection of points and not as physical objects, the paradox goes away, as the two spheres have the same number of points, 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.