# The Grey Labyrinth is a collection of puzzles, riddles, mind games, paradoxes and other intellectually challenging diversions. Related topics: puzzle games, logic puzzles, lateral thinking puzzles, philosophy, mind benders, brain teasers, word problems, conundrums, 3d puzzles, spatial reasoning, intelligence tests, mathematical diversions, paradoxes, physics problems, reasoning, math, science.

Message body

 Emoticons View more Emoticons
Options
HTML is OFF
BBCode is ON
Smilies are ON
 Disable BBCode in this post Disable Smilies in this post

 All times are GMT
 Jump to: Select a forum Puzzles and Games----------------Grey Labyrinth PuzzlesVisitor Submitted PuzzlesVisitor GamesMafia Games Miscellaneous----------------Off-TopicVisitor Submitted NewsScience, Art, and CulturePoll Tournaments Administration----------------Grey Labyrinth NewsFeature Requests / Site Problems
 Topic review
Author Message
Bicho the Inhaler
Posted: Mon Aug 20, 2007 4:05 pm    Post subject: 1

Poisonium wrote:
But then again, it is a paradox, so there is no final solution.
A paradox is usually an apparent contradiction that is resolved by a closer examination. Perhaps...

- the predicate "x is the smallest number that cannot be defined in fewer than 22 syllables" doesn't define a number (which seems contradictory, but...)
- the predicate "x can be defined in fewer than 22 syllables" isn't well-defined

That doesn't seem right, does it? Surely a number either can or can't be defined in N syllables, right? But how would you formalize it mathematically? Take the predicate "x can be defined in N syllables." A formal statement of this would resemble

Quote:
There exists a syntactic formula P with length less than f(N) such that x satisfies the predicate represented by P, and for all y, if y satisfies the predicate represented by P, then y = x or y > x.

(f(N) is an appropriate function approximating lengths in syllables by corresponding formula lengths.)

The problem with that is the predicate "x satisfies the predicate represented by P." That predicate should be true if and only if P(x) is true. However, the concept of truth cannot be defined in axiomatic mathematics. There is no formally defined mathematical predicate "T(F)" that is true if and only if the syntactic formula F is true. So the above formalization attempt fails.

So I think the case is that the predicate "x can be defined in N syllables" can't be defined mathematically when N is 22 or greater. In that case, the "set of all numbers that can't be defined in fewer than 22 syllables" is formally undefined. It doesn't seem very satisfying. I'll have to think about this more...
Poisonium
Posted: Sat Aug 18, 2007 3:00 pm    Post subject: 0

Not necessarily. As this is a paradox, there are many ways of understanding it, and I see it as though that you basically conclude they can be described in less than 22 syllables.

Say for example we have found the lowest 10 numbers that cannot be described in less than 22 syllables. We run the line "the smallest integer that can't be described in fewer than 22 syllables." on the lowest of them. That way we can cross that one out. Seeing as we have just described that one, the next number can also be described, and so on and so forth. But then again, it is a paradox, so there is no final solution.

PS: The actual name of this is "Berry's Paradox".[/spoiler]
jesternl
Posted: Sat Aug 18, 2007 2:34 pm    Post subject: -1

Poisonium wrote:
jesternl wrote:
the smallest integer that can't be described in fewer than 22 syllables.

I read as though no such number exists, as that way any number could be described in less than 22 syllables. Or maybe not.

But there can be only one of those.
Poisonium
Posted: Sat Aug 18, 2007 11:30 am    Post subject: -2

jesternl wrote:
the smallest integer that can't be described in fewer than 22 syllables.

I read as though no such number exists, as that way any number could be described in less than 22 syllables. Or maybe not.
Beartalon
Posted: Fri Aug 17, 2007 10:59 pm    Post subject: -3

Number one
Number two
Number three
...
Chuck
Posted: Fri Aug 17, 2007 6:24 pm    Post subject: -4

We want descriptions of individual integers so we can tell them apart.
Beartalon
Posted: Fri Aug 17, 2007 5:32 pm    Post subject: -5

All integers can be described in two syllables: "Number."
Chuck
Posted: Fri Aug 17, 2007 4:29 pm    Post subject: -6

All of it.
CzarJ
Posted: Fri Aug 17, 2007 3:39 pm    Post subject: -7

How much wood could a woodchuck, Chuck?
Chuck
Posted: Fri Aug 17, 2007 3:24 pm    Post subject: -8

If I make a list of all the integers that I claim are not describable in fewer than 22 syllables, you can then point to the lowest number on the list and claim that it is describable in fewer than 22 syllables. But then if I remove it from the list then that definition no longer describes it, so I have a number that's not describable in fewer than 22 syllables that's not on my list, so I have to put it back. Since the 21 syllable description cannot reliably identify this number, it is not a description of it, so it must be on the list of numbers that aren't describable in fewer than 22 syllables.
jesternl
Posted: Fri Aug 17, 2007 1:39 pm    Post subject: -9

(If nothing else, I like the name )

Clearly there are integers so huge they can't be described in fewer than 22 syllables. Put them all in a big pile and consider the smallest one. It's "the smallest integer that can't be described in fewer than 22 syllables."

That phrase has 21 syllables.