Where is this wrong?

[Kuciwalker]

Or is it?

I was thinking about the brain teaser "Let n be the smallest positive integer than cannot be defined in fewer than twenty english words", to which the answer is "there is no such n, you just defined it in seventeen words", and came to this:



Let S be the set of all positive integers that cannot be defined in fewer than twenty english words.

Assume that S is non-empty.

Order S from least to greatest element. (This is doable because these are positive integers, even though I would presume that S is an infinite set.)

The first element of S can be defined in fewer than twenty english words (in fact, seventeen), by the teaser above.

The first element of S is not in S. Contradiction. S must be empty.

Therefore, there are no positive integers that cannot be defined in fewer than twenty english words.

This seems wrong. Where's the flaw in my logic?

[Whoha]

you are using sets, theres your flaw.

[OzzyKP]

If it can't be done on a simple pocket calculator then its unnecessary math.

[Kuciwalker]

Math that can't be done on that simple pocket calculator is required to make that simple pocket calculator.

[OzzyKP]

hush you.

[Ramo]

I'm not sure why you bothered to do that proof. Your conclusion is equivalent to the implication that you made in the teaser solution. If your first statement constitutes a "definition," you have a contradiction.

There is a pretty interesting question in set theory posed by Russell:
Suppose S is the set of all sets that are not an element of itself. Is S a member of ifself?

[Jaguar]

The logic is perfectly fine. I have no idea why you bothered to do that, though.

[KrazyHorse]

Even if you'd managed to make a contradiction, all you're doing is showing that naive set theory has unprovable statements.

Duh.

[Last Conformist]

Well, S is empty, so where's the problem?

[KrazyHorse]

None.

[Spiffor]

Re: Where is this wrong?

Originally posted by Kuciwalker
Therefore, there are no positive integers that cannot be defined in fewer than twenty english words.

This seems wrong. Where's the flaw in my logic?

In your riddle, you ask to define the smallest integer that can't be defined in less than 20 words. Your 17-words definition applies for that one particular number, because it defines exactly one integer among an infinity of integers.

There is no reason to believe that you can define every individual integer with your formula (you can sure define the "second smallest", the "millionth smallest", but not the "eight thousands three hundred twenty fourth smallest" in less than 20 words)

That's simply because our vocabulary doesn't follow closely the mathematical continuum. If you want to use linguistical logics in mathematics, or vice-versa, you'll have a "contradiction" simply because the logics are different.

[KrazyHorse]

In your riddle, you ask to define the smallest integer that can't be defined in less than 20 words. Your 17-words definition applies for that one particular number, because it defines exactly one integer among an infinity of integers.

There is no reason to believe that you can define every individual integer with your formula (you can sure define the "second smallest", the "millionth smallest", but not the "eight thousands three hundred twenty fourth smallest" in less than 20 words)

WTF? I don't think you're thinking this through, spiff.

There can't be a second-smallest number without there being a smallest.

And it's been demonstrated that there is no smallest number.

The relevant fact is that any subset of the positive integers is closed and bounded below, thus it has a minimum.

[Spiffor]

Yet there are numbers that cannot be defined in fewer than 20 words, which is proof that logics sucks

[KrazyHorse]

Originally posted by Spiffor
Yet there are numbers that cannot be defined in fewer than 20 words, which is proof that logics sucks

No, it's proof that naive set theory sucks.

[KrazyHorse]

The point is not that you can define every single positive integer with 20 words or less (as a matter of fact, I can demonstrate that you can't), but that natural or naive mathematics contains Godel statements. Which Russell and Whitehead knew about way back in 1900.