Talk:Berry paradox

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[edit] 2003

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)

[edit] 2004

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)


[edit] 2005

[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] 2006

[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)

[edit] Continuing the 2004 discussion

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] 2007

In the above discussion it was said that "The point is that you have to name the numbers so that they are uniquely recognisable." In the Journal of Symbolic Logic, Vol. 53, No. 4, 1220-1223. Dec., 1988, in "The False Assumption Underlying Berry's Paradox," James D. French demonstrated that an infinite number of numbers could be uniquely described in the exact same words. French, 04 January 2007

[edit] Alternative explanation

Added in an alternative explanation of Berry's paradox. I feel the explanation using "set A" could be a bit confusing for people who do not have much knowledge of mathematics.

Saurabhb 23:09, 26 February 2007 (UTC)

Unfortunately, it's not that simple. Your "alternative explanation" explains why nothing can match the description "The smallest positive integer not definable in under eleven words", but that's hardly a paradox; there are many hypothetical descriptions that nothing matches, such as "The smallest positive integer less than zero", and most cause no consternation. What makes the Berry paradox a paradox is that something must match that description, yet nothing can. I really don't think there's a way to explain the full paradox without making reference to naive set theory, but you're welcome to give it a shot. —RuakhTALK 04:57, 27 February 2007 (UTC)
The assumption that "something must match that description" is a false assumption as demonstated by the paradox itself. WAS 4.250 16:33, 11 May 2007 (UTC)

[edit] Missing the point

(Sorry if someone has already explained this but) I think that some of you have missed the point. you do not need to go as far as 10^999999999999999 (as in one of the above examples). The paradox explicitly states the use of 'words' when searching for the number. Here is a number "One Hundred and twenty one thousand one hundred and twenty one" this is a number that is not definable in under 11 words (it requires 11 words). I am sure that there will be a smaller number that also requires 11 words or more, find it!

Once you have found that number, you (in theory) have found The smallest positive integer not definable in under eleven words.

The paradox occurs by the fact that though the number could not be defined in under 11 words, it can be defined in under 11 words by refering to the number as "The smallest positive integer not definable in under eleven words" (which defines it in under 11 words, because the phrase is only 10 words long)!

—The preceding unsigned comment was added by Cs1kh (talk • contribs) 12:28, 19 July 2007 (UTC).


[edit] Semantics

"It is generally accepted that the Berry paradox results from interpreting sets of possibly self-referential expressions: it and similar paradoxes embody so-called "vicious-circle" fallacies. To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them"

It is my belief that the entire mess lies simply in a weak definition or even misuse of the word "word". I do not see why a language must or should contain a finite number of words, only a finite number that can be expressed. It is, however, so with primes, though that is no reason to discount their multitude. While the above discussion makes the point that one might call a number "Joe" and in so doing define it, such nomenclature is really quite unnecessary, for numbers in themselves are words. Eight billion four thousand sixty two and nine millionths may very easily be considered a single word despite the spaces, just as The United States of America is oft considered a single word. With this is mind the only real constraints on naming numbers are their ability to be expressed and their system of classification. The paradox did say "can be" expressed, and so the first constraint isn't a problem at all as it would be assumed that a finite word could be expressed given enough time. The second constraint is classification. We must construct a system that both does not break down with degrees of magnitude and can express any number finitely that the previous constraint be non-constraining. (talk ) —The preceding signed but undated comment was added at 20:05, August 23, 2007 (UTC).

Just a note: the example given in the wiki-page ("The smallest positive integer"...) explicitly states/requests 'integer'. can you re-explain using only integer(s) instead of subsections/fractions (e.g. millionths) ([[User:Cs1kh]] (talk) 13:18, 7 February 2008 (UTC))

[edit] Explanation of the paradox

The paragraph explaining the paradox is rather vague; by the time that it points out that the defining phrase is itself less than eleven words long, the reader is already six lines into the explanation. This needs rewording for clarity. -- Sasuke Sarutobi (talk) 17:39, 23 May 2008 (UTC)