Introduction to paradoxes

A mathematical paradox is any statement (or a set of statements) that seems to contradict itself (or each other) while simultaneously seeming completely logical. Paradox (at least mathematical paradox) is only a wrong statement that seems right because of lack of essential logic or information or application of logic to a situation where it is not applicable. There are many paradoxes in mathematics.

There are many proofs that use proof by contradiction, where you make a statement and then prove that it is wrong by producing a contradiction.

Prove that there are infinitely many prime numbers.

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Suppose there are finitely many primes such that the product of primes can be represented as

\[2 \times\ 3 \times\ 5 \times\ \cdots \times\ p_n. \]

Now we add 1 to this and call the positive integer \(M:\)

\[M = (2 \times\ 3 \times\ 5 \times\ \cdots \times\ p_n \ ) + 1. \]

Now by the fundamental theorem of arithmetic, there must be a prime \((\)say \(p_{I}\) for \(1 < I < n+1)\) that divides both \(M\) and the product of primes, which implies that

\[p_{I} \Big| M - (2 \times\ 3 \times\ 5 \times\ \cdots \times\ p_n \ ) = 1 .\]

But no primes divide 1, so by contradiction our original statement was false. Therefore, there are infinitely many primes. \(_\square\)

The Painter's Paradox (a.k.a. Gabriel's Horn):

Using the surface and volume of revolution of a 3D shape for \( y = \frac{1}{x}\), we have

\[\begin{align} V &= \pi \int_{1}^{\infty} \big(f(x)\big)^2 dx \\ A &= 2 \pi \int_{1}^{\infty} f(x) \sqrt{1 + \big(f(x)\big)^2} dx \\ \Rightarrow V &= \pi \int_{1}^{\infty} \frac{1}{x^2} dx \\ &= \pi \left[-\frac{1}{x} \right]_{1}^{\infty} = -\pi\left( \frac{1}{\infty} -1\right) = \pi. \end{align} \]

However, we get a surprising result for the surface area:

\[\begin{align} A &= 2 \pi \int_{1}^{\infty} \frac{1}{x} \sqrt{1 + \left(\frac{1}{x}\right)^2} \\ &> 2 \pi \int_{1}^{\infty} \frac{1}{x} \\ &= 2 \pi \big[\ln x\big]_{1}^{\infty} \\ &= \infty. \end{align}\]

This implies that it is possible to fill the container, without completely covering the surface area, which creates a paradox. Don't lose hope in maths though--there is an explanation! When we are applying maths, we need to be careful about how we go about our business because there are more limitations in reality. In this case, there is actually no paradox because in maths we assume that everything can be made smaller without limit (if we are dealing with real numbers, rather than integers), which implies that the paint can be made "infinitely thin" to cover an infinite surface area. It is just like when I put \(\frac{1}{\infty}\) = 0, when in really it is more correct to put \(\lim_{x \rightarrow\infty} \frac{1}{x}\)\(\rightarrow\) 0 because we can never reach infinity and \(\frac{1}{x} > 0 \ (\forall x \in \Re , x > 0) \).

In a TV programme "Into The Universe With Stephen Hawking" on Discovery Channel, Stephen Hawking described the following paradoxical condition:

"Suppose you have somehow built a time machine. Now you open up a kit containing parts of a revolver. You assemble all the parts and then load the revolver with bullets. Now you go into past, 5 minutes back (the time when you have actually not set up your revolver) via the time machine and you shoot yourself (the man that belongs to your past who was kind of going to assemble the revolver) using the same revolver."

Here comes the twist. First of all, if the revolver has actually not been assembled, then with what have you killed yourself, and the second part is, "is it really possible to kill yourself in the past, if a time machine would have existed?"

This is in fact known as grandfather's paradox. By definition, it is described as follows:

"Suppose a time traveller goes back in time and kills his grandfather before his grandfather met his grandmother. As a result, the time traveller is never born. But, if he was never born, then he is unable to travel through time and kill his grandfather, which means the traveller would then be born after all, and thus we arrive at a contradiction and are ‘stuck’ in this ‘loop’.

It was first described by the science fiction writer Nathaniel Schachner in his short story Ancestral Voices and by René Barjavel in his 1943 book Le Voyageur Imprudent (Future Times Three).

Proposed Resolutions:

The model of time travel proposed by Seth Lloyd, et al., in a recent paper at arXiv.org arises from their investigation of the quantum mechanics of closed timelike curves (CTCs) and search for a theory of gravity. In simple terms, a CTC is a path of spacetime that returns to its starting point. The existence of CTCs is allowed by Einstein’s general relativity, although it was Gödel who first discovered them. As with other implications of his theories, Einstein was a bit disturbed by CTCs.

Einstein’s theory of general relativity implicitly allows travel to the past. It took several decades before Gödel proposed an explicit space-time geometry containing closed timelike curves (CTCs). The Gödel universe consists of a cloud of swirling dust, of sufficient gravitational power to support closed timelike curves. Later, it was realized that closed timelike curves are a generic feature of highly curved, rotating spacetimes: the Kerr solution for a rotating black hole contains closed timelike curves within the black hole horizon; and massive rapidly rotating cylinders typically are associated with closed timelike curves [2, 8, 12]. The topic of closed timelike curves in general relativity continues to inspire debate: Hawking’s chronology protection postulate, for example, suggests that the conditions needed to create closed timelike curves cannot arise in any physically realizable spacetime.

Let A be the set of all sets which do not contain themselves: \( \{S | S \not \in S\}\).

\( {1} \in \big\{ \{1\}, \{1, 2\}\big\},\) but \(\{1\} \not \in \{1\}.\)

Is \(A \in A?\)

Suppose \(A \in A.\) Then by definition of \(A,\) \(A \not \in A.\)
Suppose \(A \not \in A.\) Then by definition of \(A,\) \(A \in A.\)

Thus, we need axioms in order to create mathematical objects.

A further exploration of axiomatic set theory can be found in Principia Mathematica by Alfred North Whitehead and Bertrand Russell.

This is one of my favourite paradoxes. Zeno of Elea (c. 450 BCE) is credited with creating several famous paradoxes, and perhaps the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer’s The Iliad.) I'm presenting it in a readable dialogue format.

The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow.

“How big a head start do you need?” he asked the Tortoise with a smile.

“Ten meters,” the latter replied.

Achilles laughed louder than ever. “You will surely lose, my friend, in that case,” he told the Tortoise, “but let us race, if you wish it.”

“On the contrary,” said the Tortoise, “I will win, and I can prove it to you by a simple argument.”

“Go on then,” Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.

“Suppose,” began the Tortoise, “that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?”

“Very quickly,” Achilles affirmed.

“And in that time, how far should I have gone, do you think?”

“Perhaps a meter—no more,” said Achilles after a moment’s thought.

“Very well,” replied the Tortoise, “so now there is a meter between us. And you would catch up that distance very quickly?”

“Very quickly indeed!”

“And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?”

“Ye-es,” said Achilles slowly.

“And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance,” the Tortoise continued smoothly.

Achilles said nothing. “And so you see, in each moment you must be catching up the distance between us, and yet I—at the same time—will be adding a new distance, however small, for you to catch up again.”

“Indeed, it must be so,” said Achilles wearily.

“And so you can never catch up,” the Tortoise concluded sympathetically.

“You are right, as always,” said Achilles sadly—and conceded the race.

Zeno’s paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance…and so on forever. The consequence is that I can never get to the other side of the room and that no matter how fast or far ago, there is always a half of that number.

What this actually does is to make all motion impossible, for before I can cover half the distance I must cover half of half the distance, and before I can do that I must cover half of half of half of the distance, and so on, so that in reality I can never move any distance at all, because doing so involves moving an infinite number of small intermediate distances first.

Now, since motion obviously is possible, the question arises, what is wrong with Zeno? What is the "flaw in the logic"? If you are giving the matter your full attention, it should begin to make you squirm a bit, for on its face the logic of the situation seems unassailable. You shouldn’t be able to cross the room, and the Tortoise should win the race! Yet we know better. Hmm......

Rather than tackle Zeno head-on, let us pause to notice something remarkable. Suppose we take Zeno’s paradox at face value for the moment, and agree with him that before I can walk a mile I must first walk a half-mile. And before I can walk the remaining half-mile I must first cover half of it, that is, a quarter-mile, and then an eighth-mile, and then a sixteenth-mile, and then a thirty-secondth-mile, and so on. Well, if I could cover all these infinite number of small distances, how far should I have walked? One mile! In other words,

\[1=\frac12+\frac14+\frac18+\frac1{16}+\frac1{32}+\cdots. \]

At first, this may seem impossible: adding up an infinite number of positive distances should give an infinite distance for the sum. But it doesn’t—in this case it gives a finite sum; indeed, all these distances add up to 1! A little reflection will reveal that this isn’t so strange after all: if I can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give me back the finite distance I started with. (An infinite sum such as the one above is known in mathematics as an infinite series; when such a sum adds up to a finite number, we say that the series is summable.)

Now, the resolution to Zeno’s paradox is easy. Obviously, it will take me some fixed time to cross half the distance to the other side of the room, say 2 seconds. How long will it take to cross half the remaining distance? Half as long—only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second, and so on. And once I have covered all the infinitely many sub-distances and added up all the times it took to traverse them? Only 4 seconds, and here I am, on the other side of the room after all.

And poor old Achilles would have won his race!!

There is a very poor, quaint, little town where everyone is in a huge debt with someone but has no money to pay it back.

There is a hotel which is hardly seeing any business anymore. They are to soon shut it down. One day a very wealthy American guest shows up and wants to spend a night there. However, before he confirms, he asks for a tour of the hotel. The receptionist asks for a security deposit which the American can take back in case he doesn’t like the rooms. The guest obliges. It turns out by a matter of luck this is the exact amount that the hotel owed to the chef as salary for three months which they hadn’t been able to pay. They gave the cash to the chef.

The chef saw that this was the exact amount of cash he owed the grocer for months of groceries he hadn’t been able to pay for. He paid the grocer. The grocer realized it was the exact amount he owed the doctor for treating his wife’s arthritis, so he paid the doctor. Then the doctor paid the money to the nurse for two months of service he couldn’t pay for. The nurse was new to the town, so she had been staying in the hotel for a few days before she found a house to rent. She too was poor and couldn’t pay the hotel at that time. The money she received from the doctor was exactly what she owed the hotel, so she paid.

Now the hotel had got back the exact amount it had paid the chef. Then the guest has finished his tour of the rooms, and it turns out he doesn’t like it. He takes back his security deposit from the hotel and leaves, never to be seen again. So everyone's debt has been paid, but nothing is different from before. No one has earned anything. But now everyone is happy even though the money has done nothing but to return to its first owner. Therefore, this paradox is basically describing how nothing had changed from the start, but along the way, debts were paid, with no money.

Did the debt really exist at all?