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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!

Note by Ameya Salankar
2 years, 9 months ago

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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 · 2 years, 7 months ago

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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 · 2 years, 7 months ago

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@Krishna Ar That was pretty interesting :) Agnishom Chattopadhyay · 2 years, 7 months ago

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Nice proof! Paramjit Singh · 2 years, 7 months ago

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what about the 2nd unintresting number??????????? :P Lokesh Naani · 2 years, 7 months ago

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@Lokesh Naani Its interesting for being the 2nd uninteresting number :D Shreya R · 2 years, 7 months ago

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@Shreya R @Shreya R quite true! Ameya Salankar · 2 years, 7 months ago

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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 · 2 years, 7 months ago

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@Finn Hulse @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 · 2 years, 7 months ago

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@Ameya Salankar 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 · 2 years, 7 months ago

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@Mursalin Habib Yes,that's absolutely right. Krishna Ar · 2 years, 7 months ago

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@Mursalin Habib OK! Looks like we have found a fallacy to the proof! Ameya Salankar · 2 years, 7 months ago

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@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 · 2 years, 7 months ago

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@Mursalin Habib @Mursalin Habib, I knew this one. But there is something wrong with the proof. (The uninteresting number one) Ameya Salankar · 2 years, 7 months ago

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@Ameya Salankar What if it is an empty set? Then... :O Finn Hulse · 2 years, 7 months ago

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@Finn Hulse @Finn Hulse, I forgot that one ... :D Ameya Salankar · 2 years, 7 months ago

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@Ameya Salankar Yup...otherwise you're defying well ordering principle!!! Krishna Ar · 2 years, 7 months ago

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@Ameya Salankar Hi-fi! We have the same number of followers! :D ^_^ Krishna Ar · 2 years, 7 months ago

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@Krishna Ar @Krishna Ar, you have caught up! Initially, you had less number of followers. Great job! Keep it up! Ameya Salankar · 2 years, 7 months ago

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@Ameya Salankar Thank you ^_^. Initially I had way too less number of followers, yes now I have reached a decent level :D Krishna Ar · 2 years, 7 months ago

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