# 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.

## Famous Proofs by Contradiction

## Prove that there are infinitely many prime numbers.

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 \[p_{I} | M - (2 \times\ 3 \times\ 5 \times\ \cdots \times\ p_n \ ) = 1 .\] But no primes divide 1, so by contradiction our original statement was false. Therefore, there are infinitely many primes. \(_\square\)

## Geometry Pardoxes

The painter's paradox (a.k.a Gabriel's Horn):

Using the surface and volume of revolution of a 3D shape (of revolution) for \( y = \frac{1}{x}\), we have \[\begin{align} V &= \pi \int_{1}^{\infty} (f(x))^2 dx \\ A &= 2 \pi \int_{1}^{\infty} f(x) \sqrt{1 + (f(x))^2} dx \\ \Rightarrow V &= \pi \int_{1}^{\infty} \frac{1}{x^2} dx \\ &= \pi \left[-\frac{1}{x} \right]_{1}^{\infty} = -\pi\left( \frac{1}{\infty} -1\right) = \pi. \end{align} \] However, we get a surprising result for the surface area: \[\begin{align} A &= 2 \pi \int_{1}^{\infty} \frac{1}{x} \sqrt{1 + \left(\frac{1}{x}\right)^2} \\ &> 2 \pi \int_{1}^{\infty} \frac{1}{x} \\ &= 2 \pi [\ln x]_{1}^{\infty} \\ &= \infty. \end{align}\] 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 really it is more correct to put \(\lim_{x \rightarrow\infty} \frac{1}{x}\)\(\rightarrow\) 0 because we can never reach infinity and \(\frac{1}{x} > 0 \ (\forall x \in \Re , x > 0) \).

## Paradox of Time Travel

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 setup your revolver) via the time machine and you shoot yourself (the man which belongs to your past who was kinda going to assemble the revolver) using the same revolver."

Here comes the twist. First of all, when 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."

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.

The Kraus decomposition of a generic quantum evolution (that can describe the evolution of both isolated and open systems). It is given by L[ρ] = TrE [U(ρ ⊗ |eihe|)U †] = Xi hi|U|ei ρ he|U† |ii = Xi BiρB† i,

where |ei is the initial state of the environment (or, equivalently, of a putative abstract purification space) U is the unitary operator governing the interaction between system initially in the state ρ and environment, and Bi ≡ hi|U|ei is the Kraus operator. In contrast, the nonlinear evolution of our post-selected teleportation scheme is given by N [ρ] = TrEE′ h(U ⊗ 11E′ ) (ρ ⊗ |ΨihΨ|) U † ⊗ 11E) × (11 ⊗ |ΨihΨ|) i = X l,j hl|U|li ρ hj|U†|ji = CρC†,

where C ≡TrCT C[U] and |Ψi ∝ P i |iiE|iiE′ (or any other maximally entangled state, which would give the same result). Obviously, the evolution in (8) is nonlinear (because of the post-selection), so one has to renormalize the final state: N [ρ] → N [ρ]/TrN [ρ]. In other words, according to our approach, a chronology-respecting system in a state ρ that interacts with a CTC using a unitary U will undergo the transformation.

## Russell's Paradox

Let A be the set of all sets which do not contain themselves: {S | S 6∈ S}.

{1} ∈ {{1}, {1, 2}}, but {1} 6∈ {1}.

Is A ∈ A?

Suppose A ∈ A. Then by definition of A, A 6∈ A.

Suppose A 6∈ A. Then by definition of A, A ∈ 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.

**Cite as:**Introduction to paradoxes.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/introduction-to-paradoxes/