In a recent talk at MIT, Umesh Vazirani appealed to the famous Birthday Paradox to say that two random subsets of {1,…,N}, each of size o(√N), probably wouldn’t intersect each other. Of course we all understood what he meant, but it occurred to me that Umesh was actually appealing to the Birthday Unparadox: “If you put three people in a room, chances are no two of them will have the same birthday.”

Once I realized that, I started seeing unparadoxes everywhere I looked:

The Banach-Tarski Unparadox: If you cut an orange into five pieces using a standard knife, then put them back together, the result will have exactly the same volume as the original orange.

Braess’ Unparadox: If you add an extra lane to a highway, one possible result will be to decrease congestion.

Hempel’s Unparadox: If you observe a bunch of ravens and find that all of them are black, this might increase your likelihood for the statement “All ravens are black.”

Russell’s Unparadox: The set of all sets that contain themselves as a member, might or might not contain itself as a member (either way is fine).

In the spirit of my highly-successful Best Umeshism and Best Anthropicism contests (remember those?), I now open the floor to you: come up with the best unparadox! The winner will receive absolutely nothing. (If you have to ask what the point is, this contest isn’t for you.)

### 110 Responses to “Unparadox Contest”

1. onymous Says:

Hilbert’s hotel unparadox: if a finite number of guests show up at an infinite hotel, they can be accommodated.

Horse unparadox: suppose any pair of horses is the same color. Then all horses are the same color.

2. Ben L Says:

The famous twins unparadox: two people born on the same day will, many years later end up being the same age.

Assuming, I guess, that neither of them is dead.

3. TGM Says:

1)
The following statement is false.
The previous statement was false.

[or, replace false with true above for a free bonus unparadox]

2) This is a pipe

4. Ju Says:

The Schrödinger Unparadox : If you put a cat in a closed box full of hydrocyanic acid, he may be one hundred percent dead.

5. matt Says:

The Groucho Unparadox: I joined that club because if they really knew me, they wouldn’t have me as a member.

6. Dan Says:

The unparadox of the surprise examination: A teacher decides to have a surprise quiz within the next week, but does not tell his students. All of them are surprised by the quiz.

The unparadox of omniscience: God won’t bother playing Chicken with you unless he knows in advance that you’re going to swerve (unless he *wants* to lose).

7. Mark Gritter Says:

Skolem’s Unparadox: A formalized model of ZF set theory has a model whose universe is denumerable. Thus, objects which to us would appear to be denumerable are, in the model, also denumerable.

8. Gabe Durazo Says:

If 2+2 = 4, then I am not the pope.

9. JeffE Says:

Any countably infinite set of real numbers contains at least one real number.

The smallest positive integer describable in one word is “one”.

If you double your bet whenever you lose, you should expect to run out of money.

If there’s a goat behind every door, it doesn’t matter whether or not you change your mind.

If you drive your car at the speed of light, you won’t have enough energy left to turn on your headlights.

Scott Aaronson can consistently believe that this statement is true. (But he probably won’t.)

10. Anonymous Says:

Euler’s other celebrated equation: e^{ i pi 0} = 1

11. j.edwards Says:

Burali-Forti unparadox: The collection of all the ordinals less than a given ordinal is itself an ordinal. Also, there’s an ordinal bigger than it.

12. Lev R. Says:

I see the point of this contest, but I’m afraid that if I win, I won’t get my nothing.

13. Anonymous Says:

I think, therefore I think I am.

14. Joe Says:

The Monty-Hall unparadox: There are 3 doors with one containing a prize. You choose one of the doors. Monty asks whether you want to switch, but doesn’t open any of the doors. It doesn’t matter if you do or not.

Zeno’s unparadox: Achilles has a head start over the tortoise. The tortoise will never catch up.

The 2-envelope unparadox: I give you 2 envelopes containing the same amount of money. You choose one. Expected return doesn’t change if you then switch your choice.

15. Tobias Osborne Says:

If a tree falls down in a forest and someone is around then it makes a sound.

Oh, and what about the other Nez Naok:

What is the sound of two hands clapping?

16. anonymous Says:

A monkey typing randomly on a typewriter without the ability to change pages, will produce a small amount of giberish.

Given 6,000 (or 5758) years, natural selection will not produce organisms capble of directing the monty python sketch: “The Ministery of Silly Walks”

17. Mark Gritter Says:

Drunkard’s Unparadox: A finite random walk on the integral points in the plane will cover only a bounded portion of the space. Alternately, a finite random walk on the integral points in the plane will probably leave the drunkard as lost as before.

18. Perry E. Metzger Says:

The Weierstrass unfunction unparadox: there exist smooth continuous functions which are differentiable across their entire domain.

19. jo Says:

Petronius’ Unparadox: “Immoderacy in all things, including immoderacy.”

20. Tobias Osborne Says:

St Petersburg unparadox: a lottery game with a finite expected payoff is not worth an infinite amount of money

21. ThePolynomial Says:

An omnipotent being could make a square square. An omnipotent being could also make an object so heavy most people couldn’t lift it.

22. Khoth Says:

Time-travel unparadox: When you go back in time and don’t kill your grandfather.

Obler’s unparadox: The sky isn’t dark during the day.

An irresistible force can move a moveable object.

23. James Says:

Voting unparadox: Your vote really won’t ever change the outcome of any significant election.

24. Chris Says:

If you dump out a truckfull of sand, you get a heap of sand.

25. Chris Says:

Benford’s unparadox: in real-world data, all trailing digits are roughly equally likely.

No-shortcuts unparadox: In Manhattan, the shortest distance from point A to point B is the L_1 distance from point A to point B.

26. anonymous Says:

no cretans are liars

27. Arnab Says:

One way to create unparadoxes is by flipping quantifiers. For example:

There exists a univariate polynomial p with complex coefficients such that for all complex numbers x, p(x) = 0.

There exists a natural number k such that the primes contain an arithmetic progression of length k.

For every language L in NP, there exists a language L’ in NP such that if L can be decided in polynomial time, then L’ can also be decided in polynomial time.

For every 3-regular graph on n vertices, there exists a subset S of vertices of size less than n/2 such that the edge boundary of S is of size at least 3/4 |S|.

28. James Says:

I was told by the owner of Olga’s Penson in Rethymno, Crete, that if a Cretan says “all Cretans are liars” he is just trying to confuse you, and probably wants to sell you something.

29. Job Says:

Scott, i was looking over your Umeshisms and i don’t think they’re valid. A valid Umeshism in my view would be for example:

“If you’ve never failed an exam, then you’re spending too much time in classrooms”.

I think Umeshisms should be of the format:
“If you’ve never var(missed) {my-something} then you’re spending too much time doing f({general-something}).”

Where var(missed) is a variation of “missed” that can have the same meaning, {my-something} is some event particular to you (such as missing your own flight), {general-something} is the same event in general (such as missing everyone’s flights), and f({general-something}) is a side-effect of never doing {general-something} (such as spending too much time in airports due to never having missed anybody’s flight).

30. froystig Says:

Epimenides unparadox: A Cretan lies, “All Cretans are liars.”

And on a more serious note:
Quempel’s unparadox: “yields falsehood when appended to its own quotation” might increase your likelihood for the statement “All ravens are black.”

31. GreedyAlgorithm Says:

If the bellhop pockets your change you will be out some cash.

32. Ari Says:

Barber of Seville Unparadox: The barber of Seville shaves every man who does not shave himself. So who shaves the barber? Nobody: she waxes.

Smale’s Unparadox: A sphere can be turned inside out if you poke a hole in it first, then sew the hole back up when you’re done.

Black Hole Information Unparadox: If you take a bunch of information and put it in a filing cabinet, then information is conserved.

33. Charlie C Says:

Alfred E. Neuman unparadox: “Although the moon is only one-eighth the size of the Earth, it is farther away.”

34. Anon I. Says:

35. astephens Says:

The Futterman-Fox-Hörnblower unparadox: broken biscuits are less satisfying than whole biscuits even though whole biscuits are more satisfying once they are broken.

36. harrison Says:

Newcomb’s unparadox: If an omniscient, omnipotent being puts $1,000,000 in each of two boxes and gives you the option of taking either one or both boxes, your best strategy is to take both boxes. Drinker unparadox: If everyone in a bar is drinking, then someone in the bar is drinking. Lob’s unparadox: If this sentence is true, then this sentence is not false. 37. Morgan Says: Wilde’s Unparadox: I can resist anything except free will. 38. Jamie McCarthy Says: If you want peace, you must not go to war. Without law and order, man has no Fritos. 39. Steve Demuth Says: Academics have too little to do. 40. TGM Says: It’s better to be rich and healthy, than poor and sick. Nothing seems to motivate people more than a 25$ price, judging by the responses to Scott’s last two blog entries.

41. Joel Says:

Fermi’s Unparadox : given the state of our current knowledge, there is a perfectly plausible choice of parameter values for Drake’s equation that implies that the number of advanced technological civilizations in the galaxy about one.

42. Sam Says:

The Tristram Shandy unparadox: Tristram Shandy lived for a while, during which time he wrote an autobiography.

43. Tom Says:

Second time-travel unparadox: If you kill your grandfather today, you will still have been born.

Anna Karenina unparadox: There are no happy families. (Well, at most one.)

44. Bishop's Guest Says:

Studdies show that research causes cancer in mice.

45. Anonymous Says:

The sorites unparadox: If you add one more grain of sand to a heap, it definitely remains a heap.

The ship of Theseus unparadox: If you never replace any part of Theseus’s ship, then you are still entitled to call it by that name.

46. anonymous Says:

ceci est une pipe

47. Frank Says:

string theory unparadox: if string theory is unfalsifiable, it is not physics

quantum foundational unparadox: if different interpretations of quantum mechanics lead to the same overall experimental predictions, they are equivalent

48. Not even right Says:

What is the meaning of this unparadox contest?

My example is that every mother of a human being is a woman! Is it nonsense?

Kripke’s total lack of a paradox:
A) Saul Kripke didn’t address the truth of the sentence “This sentence is true” in his 1975 “Outline of a Theory of Truth”.
B) (A) is not true.

50. Osias Says:

An omnipotent being could also make a stone so light such an omnipotent being could lift.

51. Osias Says:

52. Osias Says:

If a man steals bread and rapes a woman, society will consider he is a thief and a rapist and morally condemn him.

53. Gil Says:

The horisontal line in the picture above looks smaller than the horisontal line in the picture below

>—————————————————

54. Gil Says:

>—>

The horisontal line in the picture above looks smaller than the horisontal line in the picture below

>—————————————————>

55. Stas Says:

If you can solve the halting problem for your Turing machine, it is not universal.

56. Bilsko Says:

Shooting and arrow at a cat in a box – the arrow might skewer the cat, it might not. It all depends on if the arrow is in motion or not.

57. Johan Richter Says:

Schrödinger’s quantum cat unparadox 2: When you open the box, the cat is either dead or alive.

58. Saurabh Says:

59. Xerxes Says:

Twin unparadox: If one twin of a pair at Fermilab takes a road trip to Milwaukee, they’ll still be pretty much the same age when she gets back.

Olber’s unparadox: If you look up into the night sky at a star, it will appear bright.

Relativistic ladder unparadox: If you ram a ladder into a closed barn door at relativistic speed, it will break.

60. Mark Says:

Schrödinger’s cat unparadox 3: If you put a cat into a box, and the box is full of poison, the cat will die.

The free-will/determinism unparadox: All events have causally sufficient preconditions. That’s why you ate a whole box of cookies even though you knew you’d feel sick.

61. Khoth Says:

Simpson’s unparadox: When two sets of data show a correlation, and when they’re combined they show the same correlation.

It’s possible to construct a Turing machine which halts if and only if it halts.

If a lightbulb is turned on for a minute, then off for half a minute, then on for a quarter of a minute, and so on, then after two minutes the switch will have been destroyed.

If an integer is picked uniformly at random from [0,1], then with probably 1 there is a Turing machine which prints that number.

62. Alex Mikunov Says:

BTW, is there any relationship between Grovers (N^1/2) algorithm and Birthday paradox (~N^1/2 of possible outcomes)

63. micah Says:

Gilbert and Sullivan unparadox: If you are not born on February 29, you will celebrate your nth birthday when you have been alive for n years.

“becomes true when preceded by its quotation” becomes true when preceded by its quotation.

Zeno’s unparadox: If I don’t fire an arrow, it will never hit the target.

64. syskill Says:

Moore’s unparadox: I believe it’s raining outside.

65. Scott Says:

is there any relationship between Grovers (N^1/2) algorithm and Birthday paradox (~N^1/2 of possible outcomes)

Good question! No.

66. Scott Says:

Academics have too little to do.

Or too much to do, thereby causing them to procrastinate in unparadoxical ways…

67. oscar Says:

If you do not send light through a screen with two slits on it, it will not make an interference pattern on the wall.

68. oscar Says:

It is possible to have a logical system which is not complete and not consistent.

69. Coin Says:

Uncatch-22:

E3.P3.2. General Criteria for Making Unfitness Determinations
E3.P3.2.1. A Service member shall be considered unfit when the evidence establishes that
the member, due to physical disability, is unable to reasonably perform the duties of his or her
office, grade, rank, or rating (hereafter called duties) to include duties during a remaining period
of Reserve obligation.
E3.P3.2.2. In making a determination of a member’s ability to so perform his/her duties, the
following criteria may be included in the assessment:
E3.P3.2.2.1. The medical condition represents a decided medical risk to the health of the
member or to the welfare of other members were the member to continue on active duty or in an
Active Reserve status.
E3.P3.2.2.2. The medical condition imposes unreasonable requirements on the military
to maintain or protect the member.
E3.P3.2.2.3. The Service member’s established duties during any remaining period of
reserve obligation…
E3.P3.6.2. Preponderance of Evidence. Findings about fitness or unfitness for Military
Service shall be made on the basis of preponderance of the evidence. Thus, if a preponderance
(that is, more than 50 percent) of the evidence indicates unfitness, a finding to that effect will be
made. If, on the other hand, a preponderance of the evidence indicates fitness, the Service
member may not be separated or retired by reason of physical disability.

Concern for one’s safety in the face of dangers that were real and immediate, is not an immediate disqualifying factor in determining the possibility of psychiatric unfitness for continued service (See DoD Instruction 1332.38)

70. Scott Says:

It is possible to have a logical system which is not complete and not consistent.

Oscar: Actually it’s not possible! If a system is not consistent then it can prove anything, therefore it’s complete.

71. Coin Says:

Scott: Well, there are paraconsistent logics, which attempt to introduce rules which allow contradictions to be dealt with without any single inconsistency exploding into trivialism; but, as a result, are even more incomplete than normal logics, since the tricks needed to contain the consequences of inconsistencies in this manner wind up significantly weakening the proving power accessible within the system.

Of course, I think one could argue this is just another kind of consistency, and the sense in which the “godel’s unparadox” above is satisfied by paraconsistent logics is just a semantic cheat…

72. oscar Says:

yes, you guys have out-geekified me.

73. anon Says:

Words describing the Tao are indeed words.

74. Daniel Gottesman Says:

The real Godel unparadox: Given any sufficiently powerful proof system, there are statements in the system which are true and yet can be proven.

75. Coin Says:

Haldane’s Undilemma: Fixing a single gene in a population in response to environmental change which makes it necessary, over a long period of time, where the environmental change is introduced gradually, and we don’t care what happens to the rest of the genome, and the gene in question offered some modest positive benefit even before the environmental change… actually isn’t really all that difficult.

76. Joshua Ball Says:

Jack Handy’s unparadox: I’d rather be rich than stupid.

If a stoppable projectile hits a movable wall, momentum is conserved.

Given two indistinguishable envelopes, one containing twice the amount of money as the other, the expected value of choosing neither is \$0.

P, Q : [2, 100]
P X Q is given to Alice, and P + Q is given to Bob.
Alice: I have no idea what your sum is.
Bob: I know that I have no idea what your product is.
Alice: …um….
Bob: What was the question?

77. Yaroslav Bulatov Says:

If you walk into a dark room and flip the lightswitch once every second, the light will be off after one minute.

Infinite sum 0+0+0+…. will stay the same after rearranging the terms.

Smallest interesting natural is “1”.

An infinitely long segment will appear flat.

Unparadox of thrift: if everyone stops working and starts spending, smaller total savings in the population will result

Berry unparadox: “The smallest positive integer definable in under ten words”

I think this entry is surely the number one.

79. MCH Says:

“If I didn’t wake up, I’d still be sleeping.”

“record will stand until it’s broken.”

“Half the lies they tell me aren’t true.”

“If you don’t know where you’re going, you’ll wind up somewhere else.”

“It ain’t over ’til it’s over.”

“You can observe a lot by watching.”

“The other team could make trouble for us if they win.”

“Steve McQueen looks good in this movie. He must have made it before he died.”

80. TGM Says:

Things are always found in the last place you look for them (if they are found at all)

81. Tom Says:

Freud’s unparadoxical quote: “Sometimes a cigar is just a cigar.”

82. Tom Says:

Old chestnut unparadox: When is a door still a door?

83. Garrett Says:

The WTF is it Unparadox; Whenever you find something… you stop looking for it. That’s why it’s always in the last place you looked.

84. Garrett Says:

Sound paradox; Listen, can you hear it?

85. Hanamaru Says:

If your grandfather is immortal then time travel is no fun.

86. Garrett Says:

Attraction unparadox; If you are attacted to something your most natural reaction is to get closer to it, and vice-versa.

87. Kulnor Says:

88. Craig Says:

Another grandfather unparadox: if your grandfather goes back in time and kills you, he will still have been born.

89. Joseph Hertzlinger Says:

Relativistic clock unparadox: A clock is placed on a rotating table. As the speed of rotation is increased the clock flies to pieces. (From A Random Walk in Science)

Goldbach’s unconjecture: Every even prime is the sum of two odd numbers. (From Gödel, Escher, Bach)

The blogging optical unillusion: If, after a sufficiently large amount of time spent reading blogs, you gradually turn your head so that you’re looking straight up, you will see … a ceiling. (From … I don’t recall the source)

90. Johan Richter Says:

Scrödinger cat unparadox nr 9999: Until you open the box you don’t know if the cat is dead or alive and might as well regard it as being in a superposition of dead and alive.

91. Matt Says:

The Second Barber Unparadox: A barber who shaves everybody shaves him (or her) self.

92. Greg Kuperberg Says:

This discussion has gotten out of hand, and it misses an important feature of the birthday unparadox as Umesh employed it: It confirms something that you expect to be true but isn’t trivial, or can be used non-trivially. In my view, that is the right criterion by which an unparadox is “good”. And we can keep the criterion that the unparadox should be a converse of some unexpected paradox.

A rather deep example: The Gödel completeness theorem, which says that every formal first-order assertion which is true, is provable.

Or, the EPR unparadox: Alice and Bob cannot send each other messages only by sharing entangled states.

Of the examples that Scott lists in his original message, none are difficult to establish, but the first three are actually important. E.g. the Banach-Tarski unparadox says that volume really does add for measurable sets, and is invariant under isometries of space.

The fourth one is silly, but it is perhaps interesting in exotic models of set theory. You can view set theory as a kind of mathematically generated “directory” structure, where elements are subdirectories and non-set elements are files. One common stricture of both set theory and file systems is that Thou Shalt Not Make Subdirectory Cycles. But there are models of set theory without this rule, in which Russell’s paradox is avoided in a different way.

93. Gil Says:

The flat earth unparadox (for those living in a flat earth): The earth looks flat because it is flat.

94. Cyan Says:

Arrow’s unparadox: Nonideal voting systems are possible.

Condorcet’s unparadox: A group of separately irrational individuals may have preferences which are irrational in the aggregate.

95. Yuval Sanders Says:

Einstein-Podolsky-Rosen Unparadox: Suppose a source emits two electrons with opposite spins, one to Alice and the other to Bob. Alice and Bob communicate and orient their bases so that Alice measures spin up always, and Bob measures spin down. Alice then rotates her measurement apparatus to measure in the |+>, |-> basis. Then she measures the electron and communicates the result to Bob. Bob finds out that she measured an electron.

96. Gil Says:

The unparadox of quantum computers: (a gesture to our host)
Quantum mechanics is correct and therefore quantum computers are realistic.

The unparadox of D-wave: (another gesture to our host)
Companies are tested by the computational complexity quality of their PR.

97. Gil Says:

I agree with Greg that there is also a serious aspect to this issue. In many cases, paradoxes and matters were common-sense fails are pushed way too far. “Braess unparadox” (If you add an extra lane to a highway, one possible result will be to decrease congestion,) is probably much more relevant to real life than the fact that sometimes adding a road increase congestion for all drivers. There are many examples of pushing too hard insights which goes against common sense. (The strongest example I am aware of is this: when my son Hagai was told at age 5 about evolution he asked my mother: “Grandma, how was it to be a monekey?”)

98. melior Says:

Fermi-Hart unparadox: Radio transmissions from other countries strongly suggest the presence of life overseas.

In Chess, either the black has a winning strategy, or the white has a winning strategy, or they both have a strategy that guarantees at least a tie.

100. ronnoc Says:

Twin Unparadox: A twin having traveled from earth to mars, at the same relative velocity as his twin on earth, will have aged the same amount, as his twin on earth, upon his return (and really forever, assuming the twins maintain the same relative velocity).

101. Andrew Says:

Einstein’s Unparadox: The surest way to outlive your enemies is to jump into a black hole.

102. Ron Says:

103. harrison Says:

The Tao Unparadox: Wei wei (doing through doing).

104. John Sidles Says:

Every system that can be understood, can be simulated efficiently.

105. Andrew Marshall Says:

From the output of all binary values on a large enough grid one may find truly surprising results, such as the story of a paradigm-shattering godsend who’s better understood by the general public than by scientists of his own field.

*(the borges attribution really belongs in fine print, on some other page)

106. Andrew Marshall Says:

PS, do the losers have to give nothing?

107. Gil Says:

(And it is not as trivial as it sounds.)

108. Ariel Gabizon Says:

If you’re not a showoff, you won’t outrun a turtle.

The easiest way to remain famous for thousands
of years is to prove something is inconsistent.

109. Ariel Gabizon Says: