Talk:Berry paradox

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However, the less formal version given above can apparently be solved by simply noting that there is nothing to stop us from giving any name we please to any number. For example, the enormous number of 10 to the power of 100 is called a googol. So, there is no smallest positive integer not nameable in under eleven words.

The above paragraph is rather missing the point. Although not saying anything untrue, it doesn't really give a resolution to the paradox. A name phrase like "the smallest positive integer not nameable in under eighteen words using words current during the nineteenth century" is also a valid example of the Berry paradox but it can't be resolved by using the "naming after the fact" method described above. -- Derek Ross 03:41 May 9, 2003 (UTC)

Hmm. But I can still name any number after a word current in the nineteenth century. There's nothing to stop us calling any number anything we please. So we can call it "Joe", or whatever. Evercat 03:45 May 9, 2003 (UTC)

Of course you can, but once you have assigned every 17 element combination of any name and any word ever spoken in any nineteenth century language including all the nonsense words that nineteenth century children invented, you will still be left with unnamed numbers, one of which will fit the paradoxical definition above. That's because the number of nineteenth century words is finite whereas the number of positive integers is infinite. And since you will have used up all the 17 element combinations of nineteenth century words, you will not be able to give the number a name consisting of less than 18 words. -- Derek Ross 03:53 May 9, 2003 (UTC)

Do I need to name every number in existence? Isn't it enough just to potentially be able to name any number that might otherwise be the Berry number? I mean, the point is supposed to be that there is no Berry number, because any number can be named with just one word. Anyway, if you're confident you're correct, feel free to revert or edit in the objection. Evercat 21:17 May 9, 2003 (UTC)

Unfortunately to use the given method to resolve the Berry paradox, you do need to be able to name every number in existence. Look at how it works. First I identify the Berry number by some slow but sure means -- let's say simple search through all the named numbers. Eventually I'll find a number which can't be named in less than 18 words (except by using the Berry phrase). Second I give it a name. Have I resolved the Berry paradox ? No, because if I repeat the process I now find that there is still a number which can't be named in less than 18 words (except by using the Berry phrase). It's a bigger number than the one I just named, of course, but one can still be found. So I give it a name. Have I resolved the Berry paradox ? Well, no. It doesn't matter how many times I name the current Berry number. There's always another bigger potential Berry number, waiting out there to replace it. To get them all, I need to be able to name every number in existence. You could say that there are an infinite number of words that can be invented and that's true but if I change the Berry sentence to something like the nineteenth century example above or, even more simply, to a form like "The smallest positive integer not nameable using less than eighty-nine ASCII characters", the naming trick won't work. -- Derek Ross 23:04 May 11, 2003 (UTC)

What's to stop us using the same name more than once? As long as we don't do so at the same time. Evercat 23:11 May 11, 2003 (UTC)
The point is that you have to name the numbers so that they are uniquely recognisable, i.e. if you name them and then explain your system to me then I should be able to tell which number a stream of words represents. Thus, naming both the numbers 10 and 12 Bob fails this criteria as if the word Bob is written down I have no way of knowing if the number is 10 or 12. -- Ams80
Anyway, here's a thought. What's the significant difference between the Berry Paradox and something like: "The atom not nameable in under 9 words" or whatever? There is no such atom, despite the huge number of them. They can all be given names. Just not at the same time. But so what? Nameability just requires the potential for that one object to be named, and every other object is irrelevant. Isn't it?
The above was written before Ams' comment. My answer to that is the same though, nameability is all about the possibility of that one thing being named. Everything is nameable. Evercat 23:20 May 11, 2003 (UTC)

Unfortunately not. Only the imaginable can be named. If you can't imagine it, you can't name it -- (and yet it may well exist). -- Derek Ross 00:27 May 12, 2003 (UTC)

Please note that the Berry Paradox emerges "naturally". Consider the very old proof that there are an infinite number of primes, and notice also that in the life of the human species only a finite number will be/have been found; call these prime' numbers. Then in an attempt to resolve this contradiction go over the traditional proof substituting prime' for prime throughout. Then note the false statement(s) in the revised proof. No doubt there are other places the Berry Paradox could emerge. PML.


However, the less formal version given above can apparently be solved by simply noting that there is nothing to stop us from giving any name we please to any number. For example, the enormous number of 10 to the power of 100 is called a googol. So, there is no smallest positive integer not nameable in under eleven words.

How is this a disproof, provided we have a language with words limited to some finite length?

Because you don't need to actually name every number - all that matters is that every number is nameable, ie can be named. Everything can be named. Not all at once, but that's irrelevant. Evercat 12:53 30 Jun 2003 (UTC)
My intuition of "named" does not extend to this. I think in terms of what might be written as "putting names into a one-to-one (or many-to-one) correspondence with named objects". Given any single naming scheme, the above does not hold water. If you can switch naming schemes on the fly, practically anything can be proved or disproved... Karada 13:08 30 Jun 2003 (UTC)
Example "one = two". On the left hand side, "one" denotes 1, and "two" denotes 2. On the right hand side, "two" denotes 1, and "one" denotes 2. "But that's silly", I hear you say, "you can't just change notation in the middle!" Exactly. -- Karada 13:21 30 Jun 2003 (UTC)

If today (14th sept. 2004) you call 10^999999999999999 "Bob", then "Bob" means 10^999999999999999 ONLY in 2004 English and (maybe) later ones, but it doesn't mean 10^999999999999999 in 19th Century English. Unless you are sure that somebody called a number "Bob" in the 19th century, you can't say that "Bob" is a 19th Century English word for any number. --Army1987 17:21, 14 Sep 2004 (UTC)

I appreciate that this debate seemingly ended ages ago, but I thought I would contribute my answer to this merely for the sake of anyone, like me, who decides to take a gander at the discussion, particularly those similarly confused with this paradox as it appears many people are.

The simple fact is that "naming numbers with random words", ie "Joe" etc as previously used for this example, is entirely beside the point. The point is that the statement we all try to disprove - in the original paradox, it is "The least integer not nameable in fewer than nineteen syllables" has been specifically designed to be a phrase which goes under the required syllable (or word, etc) count. It doesn't matter at all about any number which can be named. Consider any number which you know and can easily proove that you name in under nineteen syllables to be discounted. Now take the lowest number out of what remains. That number then becomes defined as "the least integer not nameable in fewer than nineteen syllables". But in becoming that number, it has become a number which can be defined in under nineteen syllables, so it is discounted. Thus we move onto the next-lowest. But doing this means that number can now be defined in less than nineteen syllables, so we discount it and move onto the next. This process forces us to discount every single number we come across, ad infinitum - at the end, after an infinite number of discounted numbers, we find that no number has not been discounted, and so there is no number which cannot be defined in under nineteen syllables.

Again, sorry to anyone who takes offense to or finds annoyance in my contributing to a discussion which ended so long ago, but I believed that this discussion was in need of another explanation. That, and I do enjoy the challenge of explaining something like this which others struggle on. I would edit the actual article to make it more clear, but I'm not sure I could do the article justice. Anyone else who reads this and feels up to the challenge is welcome to use anything I just wrote as part of their edit, though, if it helps. Falastur 03:16, 14 April 2006 (UTC)

[edit] MathWorld Entry

At Wolfram Research's MathWorld.com website the Berry Paradox can be found at this location: http://mathworld.wolfram.com/BerryParadox.html. It seems to not have the restriction to positive integers. Is this restriction necessary to maintain on Wikipedia?

That definition uses the phrase "least integer" - I'd contend that "least" implies quantity, which in turn implies non-negative numbers. If it had said "lowest integer" on the other hand, that would be a different matter, as there are an infinite number of very large negative numbers we can't possibly name within eleven words. I'm not sure whether this in itself invalidates the statement, but at minimum it adds an unnecessary complication to it. Slovakia 03:24, 4 December 2005 (UTC)

[edit] Another Different Naming Structure

And what about the fact that there are different ways of expressing any numnber? For instance, the number "one thousand five hundred and twenty one" could be easily be expressed as "five hundred and seven times three" - which is only 6 words, to the original 7.

Reffering to a number in this way doesn't require actually renaming the number, because this way of expressing it already existed. 65.95.245.90 06:51, 26 May 2005 (UTC)

note - the above is actually me, i just forgot to log in at the time! Oracleoftruth

Reading something like that in words, I'd be likely to understand 500+7×3=521, instead of 507×3=1521.--Army1987 22:00, 9 August 2005 (UTC)
It doesn't help you to avoid the paradox. In fact, it just serves to make you run out of names (or descriptions, or whatever you want to call them) faster. There can only be N^10 possible names, where N is the number of words in the English language, so Berry's number is some figure below N^10. Slovakia 03:24, 4 December 2005 (UTC)

[edit] 21, not 91

The smallest positive integer not nameable in under two words is 21, not 91, right? --ChadThomson 09:20, 30 November 2005 (UTC)

  • Just came on here to ask that myself. The phrase "A reasonable definition of English" apparantly reshapes the berry paradox into "The smallest positive integer not nameable in under two words, whose subsequent integers also possess this quality," which is stretching it a bit :P GeeJo (t) (c) 11:58, 2 December 2005 (UTC)
Ditto. I've decided to Be Bold and fix it (and in the process drop in an extra hint about why the two-word case is not problematic, because the statement doesn't satisfy its own criteria). Slovakia 03:24, 4 December 2005 (UTC)

The basic idea of the proof is that a proposition that holds of x if x = n for some natural number n can be called a "name" for x.

I'm taking the liberty to change this to a name for n . --62.219.170.138 08:21, 5 April 2006 (UTC)

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