The Uninteresting Number!

Every number is interesting. Take for example :

$1$ is neither a prime nor composite
$2$ is the first & the only even prime
$3$ is the first odd prime
$4$ is the first composite number

& so on.

But is there an uninteresting number? Let us find out.

Let $U$ be the set of all the uninteresting numbers.
Obviously, this set has a smallest element.

But such a number would be interesting because it is the first uninteresting number!

Hence, EVERY number is interesting!

#Proofs #JustForFun #Lol

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Obviously, this set has a smallest element.

This is true if the word 'number' means a non-negative integer in this context. The well ordering principle works for non-negative integers only.

This seemingly pradoxical result arises because 'interesting' -ness is not a well-defined property. Would a number be considered interesting if it's uninteresting? Can something have a property $A$ by not having the property $A$ ?

Mursalin Habib - 7 years ago

This is exactly how Agnishom Chattopadhyay won against me in a debate- " Are all people interesting" He was in the proposition and came up with something similar and said that being a bit uninteresting is in itself an interesting property...thus :P

Krishna Ar - 7 years ago

That was pretty interesting :)

Agnishom Chattopadhyay - 7 years ago

Nice proof!

A Brilliant Member - 7 years ago

what about the 2nd unintresting number??????????? :P

Lokesh Naani - 7 years ago

Its interesting for being the 2nd uninteresting number :D

Shreya R - 7 years ago

@Shreya R quite true!

Ameya Salankar - 7 years ago

Using your method of contradiction, you have thrown away a seemingly extraneous solution.

It is more logically sound to say that all numbers are uninteresting, because in this case, a number isn't interesting, they are all uninteresting.

You've also stated

Obviously, this set has a smallest element

which is an assumption.

So really, no numbers are interesting.

Finn Hulse - 7 years ago

@Finn Hulse,

Obviously, this set has a smallest element

is not an assumption! It's common sense!

How come a set doesn't have a smallest element?

Ameya Salankar - 7 years ago

Careful! A set doesn't have to have a smallest element. There is no smallest element in the set $S=\left\{ x : x \text{ is a country} \right\}$ .

Even if you're talking about sets of numbers, the statement is not always true.

For example: $A=\left\{ x : x \in \mathbb{R}, 2<x\leq 3\right\}$ has no smallest element.

The well ordering principle works on non-negative integers only.

Mursalin Habib - 7 years ago

@Mursalin Habib – OK! Looks like we have found a fallacy to the proof!

Ameya Salankar - 7 years ago

@Ameya Salankar – That is not the fallacy actually. It is understood that the word 'number' means non-negative integer in this context. The real issue here is that 'interesting' is not well defined.

Here's a more interesting [pun intended!] variant of this.

Is it possible to describe every non-negative integer with $15$ words or less?

Here's an example. $4294967297$ is the first Fermat number that is not a prime. This description uses less than $15$ words. Is it possible to do that for every number?

Let's assume the contrary. Then set of numbers that can not be described with $15$ words or less is non-empty.

Now take the smallest element in this set. This is the smallest number that can not be described with $15$ words or less. Wait! That is a way to describe that number! Let's see how many words we used. I'll be damned! We used exactly $15$ words! This number does not belong in this set now. In a similar manner it can be shown that no number can be in this set.

The problem with this whole argument is that 'description' is not well defined. What counts as a 'description'? Is it okay for a description to be inconsistent?

Mursalin Habib - 7 years ago

@Mursalin Habib – @Mursalin Habib, I knew this one. But there is something wrong with the proof. (The uninteresting number one)

Ameya Salankar - 7 years ago

@Mursalin Habib – Yes,that's absolutely right.

Krishna Ar - 7 years ago

Yup...otherwise you're defying well ordering principle!!!

Krishna Ar - 7 years ago

What if it is an empty set? Then... :O

Finn Hulse - 7 years ago

@Finn Hulse – @Finn Hulse, I forgot that one ... :D

Ameya Salankar - 7 years ago

Hi-fi! We have the same number of followers! :D ^_^

Krishna Ar - 7 years ago

@Krishna Ar – @Krishna Ar, you have caught up! Initially, you had less number of followers. Great job! Keep it up!

Ameya Salankar - 7 years ago

@Ameya Salankar – Thank you ^_^. Initially I had way too less number of followers, yes now I have reached a decent level :D

Krishna Ar - 7 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$