# Wikipedia commentary/Making fun of Britannica

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People often claim that Wikipedia can never be as authoritative as Britannica. There can be no question that at the current stage, EB is vastly more complete and accurate than Wikipedia. This page is here to exhibit mistakes and omissions in EB and how they have been corrected in Wikipedia.

### Transfinite numbers

The entry about "transfinite number" in EB claims that aleph-one is the cardinality of the real numbers. This is in fact neither provable nor disprovable; see cardinality and continuum hypothesis for the full story.

In "number game" under "logical paradoxes" in EB it is claimed that the statement "All Cretans are liars", if uttered by a Cretan, is self-contradictory. This is false; see Epimenides paradox for the correction. AxelBoldt sent a letter to the math editor of EB to point this out and received an answer three months later saying that his "sources" disagreed and considered the text to be correct.

### NP problems

In "NP-complete problem" you can find the statement

A problem is called NP if its solution (if one exists) can be guessed and verified in polynomial time;

The insert "(if one exists)" makes clear that the author does not understand that only decision problems belong to the class NP. Every instance of every problem in NP has a solution: it is either YES or NO. Only YES answers need to be verified quickly. See Complexity classes P and NP.

### Speed of X-rays in glass

Under "refractive index" in EB, the definition of the refractive index does not clarify the distinction between phase velocity and signal velocity; it is stated that the velocity of x-rays in glass is higher than the velocity of x-rays in vacuum. This is true for the phase velocity, but the speed with which information can be transmitted is not higher in glass than in vacuum.

### Real numbers

EB claims that the cardinal number aleph-0 is commonly added to the class of real numbers; this is not common at all. See real number.

The EB article about "Olbers' paradox" offers a resolution involving average lifetimes of stars which does not make sense. See Olber's paradox.

### Big O

"Big O" is a common notation in computer science and mathematics, used to compare the speed of growth of functions (also called "Landau's symbol"); it is also the name of a Japanese Anime show. Neither of which can be found in EB. See Big O and Big O anime.

### Leap years

EB claims in its leap year article that years divisible by 4000 may be non-leap years. This is in fact not an official rule and would not increase the calendar's accuracy. See leap year.

### Fast Fourier Transform

The Fast Fourier Transform, a fast algorithm to compute the discrete Fourier transform, has been called one of the most important algorithms ever. Neither the Fast Fourier Transform nor the discrete Fourier transform are mentioned in EB.

### Uncertainty Principle

EB has two articles about Heisenberg's Uncertainty Principle: one about the principle itself and another one inside the quantum mechanics treatment. Unfortunately, the two articles give different formulas: one uses h/2π and the other h/4π. Furthermore, they never make clear what exactly is meant by "uncertainty".

### Cauchy sequences

In the algebra article, subsection "Completion of fields", Cauchy sequences are defined as follows:

A sequence {xn} = {x1, x2, x3,...} of elements xn of F is called a Cauchy sequence (for the valuation φ) when, given ε> 0, there is an integer N such that the value of φ at the difference of two elements with subscripts sufficiently large is less than ε.

This is formally correct but unnecessarily complicated: the variable N is not needed.