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# Every Natural Number can be Unambiguously Described in Fourteen Words or Less

## Question

The claim is that any natural number can be completely and unambiguously identified in fourteen words or less. Here a "word" means an ordinary English word, such as one you might find in a dictionary.

You know this can't be true. After all, there are only finitely many words in the English language, so there are only finitely many sentences that can be built using fourteen words or less. So it can't possibly be true that every natural number can be unambiguously described by such a sentence. After all, there are infinitely many natural numbers, and only finitely many such sentences!

And yet, here's a supposed "proof" of that claim. Can you figure out what's wrong with it?

The Proof:

1. Suppose there is some natural number which cannot be unambiguously described in fourteen words or less.

2. Then there must be a smallest such number. Let's call it n.

3. But now n is "the smallest natural number that cannot be unambiguously described in fourteen words or less".

4. This is a complete and unambiguous description of n in fourteen words, contradicting the fact that n was supposed not to have such a description!

5. Since the assumption (step 1) of the existence of a natural number that cannot be unambiguously described in fourteen words or less led to a contradiction, it must be an incorrect assumption.

6. Therefore, all natural numbers can be unambiguously described in fourteen words or less!

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this sentance would areadly have been used; that's where the the falsehood of this logic lies.

let's say there are 5 "small" numbers that are pregressively more difficult to describe with 14 words or less. the third one, the only way to describe it is: "the smallest number that cannot be unabiguously be described in 14 words or less." although the statement itself is self contradictory, as your describing the number in 14 words, its a valid decription of the number. now what happens when you get to 4? or even 5?

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But now n is "the smallest natural number that cannot be unambiguously described in fourteen words or less".

That sentence is not a true description of n because it veracity would lead to a contradiction (Similar to "This statement is false"). Therefore it is an untrue statement, and thus a false description of n.

Edited by vinay.singh84
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But now n is "the smallest natural number that cannot be unambiguously described in fourteen words or less".

That sentence is not a true description of n because it veracity would lead to a contradiction (Similar to "This statement is false"). Therefore it is an untrue statement, and thus a false description of n.

It is a self-referential paradox, but perhaps of a slightly different stripe than the liar's paradox.

It describes a number truly. But in a way that is different from saying x is an integer greater than 5 and smaller than 7.

It is a linguistic description, instead. But it is unambiguous.

Of all numbers that can be spoken [like one hundred twenty-three (6 syllables)],

there is a unique smallest number that does not require 15 or more syllables to speak it.

But then using the phrase "the smallest number ... words or less" uses a different linguistic description.

The usual work-around is to disallow mixing the two description types in a single discussion.

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I might be necro-posting here, but I find this interesting. The problem with the "proof" is that it tries to pass off the English language as a formal mathematical language. By what standards do we decide if a sequence of words unambiguously describes a natural number? Are those standards semantic or syntactic? Let's break it down.

If certain sequences of words unambiguously describe natural numbers, let S be the set of those sequences. Define the function f : S → N that maps each such sequence of words to the natural number that it describes. Your claim is that f is surjective. Your "proof" is as follows:

1. Suppose f is not surjective. Then it misses a non-empty subset of N.

2. Any non-empty subset of N has a smallest element.

3. Thus we can define n as the smallest natural number that f misses.

4. But now f(the smallest natural number that f misses) = n. So f did not miss n, a contradiction.

5. Since assuming that f is not surjective leads to a contradiction, that assumption must be false.

6. Hence f is surjective.

This rewording of the OP should make it clearer. Notice how the phrase "the smallest natural number that f misses" is used semantically in step 3, but syntactically in step 4. If we did invent a formal mathematical language that allowed such syntax, then we have only used that piece of syntax once in our "proof". So we could just replace it with any other piece of unused syntax by argument of symmetry. Our language, our rules:

1. Suppose f is not surjective. Then it misses a non-empty subset of N.

2. Any non-empty subset of N has a smallest element.

3. Thus we can define n as the smallest natural number that f misses.

4. But now f(Mynd you, møøse bites Kan be pretti nasti...) = n. So f did not miss n, a contradiction.

5. Since assuming that f is not surjective leads to a contradiction, that assumption must be false.

6. Hence f is surjective.

By now it is made clear that step 4 is technically just a second assumption rather than a justified conclusion.

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