Berry's Paradox

[bonanova]

OK, the reasoning goes like this.

According to common sense and also something called the Well-ordered Principle,

any set of numbers can be ordered, least to greatest; the only exception being

the empty set. Every non-empty set of numbers has a member which is the

smallest member of that set.

Next...

Using up to N [N is finite] syllables, in various combinations / permutations,

only a finite number of numbers can be described. For example, for N=2 those

numbers would be

1, 2, 3, 4, 5, 6, 8, 9, 10, 12 [using 1 syllable] and

7, 13, 14, 15, 16, 18, 19, 20, 30, 40, 50, 60, 80, 90 [using 2 syllables]

So those are the numbers that can be described using fewer than 3 syllables.

If it could be shown that there are no numbers that cannot be described

using fewer than 3 syllables, then these would be all the numbers that exist.

A finite number. It would be like proving there are no numbers that require

3 or more syllables.

Let's see if that's possible to do. We ask, what is the smallest number that

cannot be described using fewer than 3 syllables? Well, there is an answer

to that. It's 11 - e-lev-en - 3 syllables. There are others, of course, like 17,

21, 22, ... but 11 is the smallest one.

So there are numbers beyond those describable using fewer than 3 syllables.

But now we ask, what is the smallest number that cannot be described using

fewer than 23 syllables. Well, there seems to be an answer to that as well.

It's 1,777,777. -- 23 syllables, and no one found a smaller one.

Enter the paradox.

1,777,777 was determined to be the number that is described by the phrase

the smallest number that cannot be described using fewer than 23 syllables.

But that phrase has 22 syllables. Ooops!

By that logic 1,777,777 cannot be - nor can any other number be - the smallest

number not specifiable using fewer than 23 syllables. That is, the set of numbers

described by that phrase has no smallest member. By the well-ordered principle,

therefore, that set of numbers is empty.

Now let's talk about the set of all numbers. It comprises two subsets:

[1] the set of all numbers that can be described using fewer than 23 syllables - a finite set [since 23 is finite.]

[2] the set of all numbers that cannot be described using fewer than 23 syllables - by WOP, the empty set.

Thus the set of all numbers is finite.

What have the great brains of our time done about things like this?

They note that the heart of the paradox is that it references itself.

Rather, the answer is described on one level, and disallowed by a

description on another level. That type of paradox is called self-referential.

They deal with it by assigning its statements a level, according to a hierarchy.

Then they allow a statement to reference only those objects on its own level of hierarchy.

In this case the number of syllables in speaking the number would be on a

different hierarchical level from the number of syllables in the phrase that

describes the number. That phrase would then not be permitted to disallow

the answer found by counting syllables.

Bertrand Russel once said,

The point of philosophy is to start with something so simple as not

to seem worth stating, and to end with somethiong so paradoxical

that no one will believe it.

[Guest]

Thanks bonanova. Believe it or not, that made sense.

However, the concept of infinity implies that for any integer, i, there exists i+1. Meaning there are an infinite number of integers.

Since Berry's Paradox leads the conclusion that there are only a finite number of integers, we must then conclude that the concept of "infinity" is flawed, with the following repercussions:

a) the universe's size is a bound value

B) the universe cannot keep expanding

c) there is an upper limit to the number of books Stephen King can write

d) there will be a final episode of General Hospital.

However, since we know that d is impossible, we enter an entirely new kind of paradox.

[Guest]

I think I am somehow missing something, and have an observation.

First, I get the paradox part, and did from the beginning, but I've never seen how

Thus the set of all numbers is finite.

Really, we are trying to define ALL numbers thus:

[1] the set of all numbers that can be described using fewer than 23 syllables - a finite set [since 23 is finite.]

[2] the set of all numbers that cannot be described using fewer than 23 syllables - by WOP, the empty set

Now we only found the SMALLEST of the numbers defined by #2, but there are an infinite amount that go above that number, many of which I am sure cannot be described using fewer than 23 syllables when written out in any way. For example 1,277,777,777. How do you describe this? One-Bill-ion Two-Hund-red Sev-en-ty Sev-en Mill-ion Sev-en Hund-red Sev-en-ty Sev-en Thou-sand Sev-en Hund-red Sev-en-ty Sev-en (33 syllables). It clearly fits into #2. But you CANNOT define it as

the smallest number that cannot be described using fewer than 23 syllables

Nor is it even the smallest number that cannot be described using fewer than 33 syllables. Therefore any description we use invalidates the proposition that ALL numbers contained within the two sets described above fit the paradox. Thus, the set of all numbers are not finite, even using the definition from 1 & 2 above, as I can fit an infinite number of numbers into #2.

Now for my observation. This whole premise requires use of the English language, which has absolutely nothing to do with the language of math, except for convenience of those who prefer to use the language. I am certain that if, for example, we were using Spanish, this whole concept would be moot. (My spanish isn't good enough to lay it out here). Or, if not Spanish, then any other language - even Klingon - is still just a cultural affectation so that we mortals can convey concepts to one another. Therefore no single number (number as a mathmatical concept) is dependant upon the word we attach to define it. The concept defines the word in this case, as [1] is one, uno, eins, ichi, un, etc. If a group of my friends decided from henceforth [1] would be considered "Supercalifragilisticexpialidocious" then nobody could invalidate that as being correct, at least within my circle of friends who would shout Supercalifragilisticexpialidocious with glee every time the Ace of spades appeared in our card game. So to use any language beyond the language of mathmatics to define a mathmatical concept is folly. As far as I know, you can't (without some sophistry) define the concept of "two" as the concept of "three." Sure, you can attach any modifier (or word) to the concepts, but you cannot make the concepts equal. Therefore you can't truly define "infinite sets" as a mathmatical concept using the weaknesses of a language and then assume that the words applied adequately define the concept. We must first have the concept and then make sure the words we apply adequately convey that concept.

[bonanova]

I think I am somehow missing something, and have an observation.

First, I get the paradox part, and did from the beginning, but I've never seen how

Thus the set of all numbers is finite.

Really, we are trying to define ALL numbers thus:

[1] the set of all numbers that can be described using fewer than 23 syllables - a finite set [since 23 is finite.]

[2] the set of all numbers that cannot be described using fewer than 23 syllables - by WOP, the empty set

Now we only found the SMALLEST of the numbers defined by #2,

but there are an infinite amount that go above that number,

many of which I am sure cannot be described using fewer

than 23 syllables when written out in any way.

You say that you saw the paradox, but you say we did find the smallest number .... etc.

The paradox says that we did not find ... [nor can anyone find - nor is there]

the smallest number not describable using fewer than twenty-three syllables.

Since that set lacks a smallest member it is the empty set.

[Guest]

Ah, I think I get it now. The fallicy is juxtaposing the well ordered principal and the concept of "the smallest

number not specifiable using fewer than 23 syllables" to define an empty set. Sorry I was so slow on that one.

[bonanova]

Since Berry's Paradox leads the conclusion that there are only a finite number of integers, we must then conclude that

d) there will be a final episode of General Hospital.

However, since we know that d is impossible, we enter an entirely new kind of paradox.

I read your post way too fast, and missed [d].

Fabulous...!

Now, who's gonna tell the producers?

[bonanova]

Ah, I think I get it now. The fallicy is juxtaposing the well ordered principal and the concept of "the smallest

number not specifiable using fewer than 23 syllables" to define an empty set. Sorry I was so slow on that one.

Not at all.

Re-reading my post, I don't think I made the connection clear; tossed it in as a teaser, kind of ...

[WitchOfDoubt]

Let's make it worse.

Allow me to define a new word: 'blonk.'

Blonk is a generic term for 'the smallest number not specifiable using fewer than 23 syllables.'

Have I just specified it in one syllable?

[Kikacat123]

I think that I might have an answer to this question. We were only supposed to use positive numbers, which technically, includes fractions. This is the smallest finite number that I can think of that is not specifically using fewer than 23 syllables. 1/googolplex to the googolplex power to the googolplex power...and continuing on for slightly less than infinity. This fits all requirements, but I am not sure if it is what you were looking for. It is an interesting question, though! : )

Edited April 7, 2013 by Kikacat123

[Kikacat123]

Sorry, correction for above-"not specifiable using fewer than 23 syllables".

[vigmeister]

Back here after almost 6 years of absence I used to be known as kingofpain, but I lost the password... I still remember discussions with bonanova, so it only made sense that start off replying to him

An assumption you make is that all numbers are specifiable. This is not proven.

- The list of specifiable numbers with less than 23 syllables is finite.

- "The smallest number not specifiable using fewer than 23 syllables" does not exist

The conclusion here is that if a number is specifiable, it can be specified with less than 23 syllables. In other words, the list of specifiable numbers is finite. PAradox resolved!

[iSpelBadlie]

the smallest number not specifiable using fewer than twenty-three syllables describes a number, and it is 23 syllables! there you have it.

[iSpelBadlie]

i meant 22,sorry

[iSpelBadlie]

and since you dont specify a number, it could be negative infinity for all i care (which is also less than 23 sylables)