Talk:Berry paradox

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)

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)

Uniquely recognisable

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

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. —Ruakh_TALK 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)

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)

Article's Self Critique

The article currently has a self-critique that contradicts another part of the article:

The argument that "Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words" assumes that "there must be an integer defined by this expression" which is counterfactual as most phrases "under eleven words" are ambiguous to their defining of an integer, with this ten word paradox being an example. Assuming one can match word phrases to numbers is a mistaken assumption.

As an article should be self consistent, we should address the critique and rewrite the article. (aside: Should an article not critique itself? Note boilerplates are about articles and not a part of them, and thus can critique articles.) I have a few questions and points to start discussion:

• Where the critique says "there must be an integer defined by this expression", does "this expression" refer to the statement of the Barry paradox here called B: "The smallest positive integer not definable in under eleven words"? Does it refer to the property ~D(n): "n is an integer not definable in under eleven words"? Does it refer to D(n): "n is an integer definable in under eleven words"? Some other expression?
• Why does the argument "Since there are infinitely many positive integers" assume there is an integer defined by the expression? Isn't it deducing it from the preceding statement which uses the pigeonhole principle: "Since there are finitely many words [...]"? Perhaps this first statement is making an assumption that an N exists such that D(n).

French's 2007 comment about an infinite number of uniquely described numbers applies somewhere to the critique. I'd like to hear more about it (as I haven't yet been able to obtain a copy of the original paper). More thoughts to follow as I try to wrap my head around the critique.Kanenas (talk) 18:59, 27 July 2008 (UTC)

Resolution

"The number not nameable0 in less than eleven syllables" is 16 syllables long even if the "0" is silent, or more depending on how it's pronounced, so this is completely irrelevant to the paradox.

Also, the section is highly redundant. In particular, the 4th paragraph is almost a straightforward restatement of the 1st.

I'll try to fix it, but no promises. --76.202.59.43 (talk) 04:13, 25 February 2009 (UTC)

Nonsense

In the "Resolution" section, there is the following non-sensical passage:

The argument presented above that "Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words" assumes that "there must be an integer defined by this expression". This is counterfactual, as most phrases "under eleven words" are ambiguous to their defining of an integer, with this ten word paradox being an example. Assuming one can match word phrases to numbers is a mistaken assumption.

Certainly the first expression quoted makes no assumption whatsoever regarding definability of integers by eleven-word phrases. It merely makes the observation that, of the eleven-word phrases that do define some integer uniquely, there are only finitely many. That is certainly true, since there are finitely many phrases using eleven words of the English language (of which the unambiguous -- uniquely defining -- phrases form a subset). The corollary is that, indeed, there are only finitely many numbers uniquely defined by eleven-word phrases.

In light of this, I will hereby delete the offending passage.Computationalist (talk) 18:44, 10 August 2010 (UTC)

The "Resolution" is not one

The section titled "Resolution" does anything but resolve the paradox and should be deleted until someone comes up with an adequate explanation of the paradox. Its first paragraph reads:

"The Berry paradox as formulated above arises because of systematic ambiguity in the word definable. In other formulations of the Berry paradox, such as one that instead reads: "...not nameable in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to vicious circle fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal. 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."

How about elaborating on just how the "systematic ambiguity in the word definable" leads to the paradox -- instead of suddenly changing the subject to "other formulations of the Berry paradox" ??? First deal with *this* formulation, and not by a breezy dismissal like "systematic ambiguity" that avoids any attempt to actually communicate exactly how the putative reasoning in this paradox goes off track.

Then, once that is done successfully, by all means mention other formulations and generalizations to your heart's content. But please first address the subject of the article rather than anything but.Daqu (talk) 07:35, 6 February 2011 (UTC)

Also, what do the subscripts mean in the resolution section? If it's arbitrary, can we use something more intuitive? For example, the number of characters allowed.

Assumes a one-to-one mapping

Maybe I'm missing something, but I could not find if it was stated that there has to be a one-to-one mapping from the combination of words to the integers, which is clearly what the section The paradox assumes. Since, if you first come up with the set you can normally define in less than eleven words, and then "all numbers previously not described in less than eleven words" and you've captured all of the numbers that were considered to be outside the set. If all it needs is the distinction between homomorphism and isomorphism, it's not really a paradox, you just have to set up the language correctly (not the same thing as what the Resolution section is talking about). Thus, it seems more like an ill-posed statement.

If I'm missing something important please let me know, I do not claim to be an expert on this; just curious. (Jahansen (talk) 05:41, 30 January 2014 (UTC))

You're not defining a set, you're defining an integer. Consider the text "a multiple of two which is less than pi" - this defines a single positive integer, two. But "a multiple of two which is less than two pi" gives the set {2,4,6}, which is not defining an integer. To define an integer you need to exclude all other integers. -mattbuck (Talk) 09:14, 30 January 2014 (UTC)

Rejection of argument by contradiction

I removed the following comment by an anonymous editor:

NOTE: Out of a contradiction one can prove anything. So, if you accept that "the expression is self-contradictory", you can conclude that "there can (or cannot) be any integer defined by it". What the original writer wrote, does not follow.

I'm posting it here in case anyone wants to discuss the topic. --Slashme (talk) 09:39, 29 February 2016 (UTC)

There is a proof against your argument. --5.249.127.17 (talk) 15:23, 28 July 2016 (UTC)

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Which paradox did Berry actually suggest?

The lede paragraph says:

Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G. Berry (1867–1928), a junior librarian at Oxford's Bodleian library, who had suggested the more limited paradox arising from the expression "the first undefinable ordinal".

The second part of this sentence is uncited, and I believe it is false.

In the 1908 paper Mathematical Logic as based on the theory of types, which I believe is the original printed source for the Berry paradox, Russell discusses seven related paradoxes, including what have become known as the Epimenides, Berry, Richard, and Burali-Forti paradoxes. The one he attributes to Berry is paradox number 4, of which he says, in part:

4. … Hence, “the least integer not nameable in fewer than nineteen syllables” must denote a definite integer…

(You can presumably imagine the rest.)

The sentence I quoted from our Wikipedia article, however, seems to be referring to paradox number 5:

5. Among transfinite ordinals some can be defined… the total number of possible definitions is $\aleph _{0}$… there must be undefinable ordinals, and among these there must be a least.

(Again you can imagine how it proceeds.)

Russell does not attribute paradox #5 to Berry. Instead, he provides references to discussions in papers of König, Dixon, and Hobson from 1905–1906.

So contrary to the claim of our article, it appears that, at least according to Russell, the Berry paradox is not “the more limited paradox arising from the expression ‘the first undefinable ordinal’”. That more limited paradox (#5) had already been considered by several mathematicians. What Russell specifically attributes to Berry, paradox #4, is what is now known as the Berry paradox itself.

If there is no substantive objection, I will correct the lede paragaph shortly. —Mark Dominus (talk) 21:06, 2 August 2017 (UTC)

Which theorem required the ordinal to exist?

In the last paragraph of the overview section, there is the following quote:

"...According to Cantor’s theory such an ordinal must exist, but we’ve just named it in a finite number of words, which is a contradiction."

It is not stated what specifically "Cantor's theory" refers to, nor have I been able to find what it could refer to, does anyone know which theory/theorem this refers to specifically? Cheers --BestEye0 (talk) 08:30, 4 June 2023 (UTC)