Where is this wrong?

[KrazyHorse]

The problem, of course, is in the omega-completeness...

Or induction, in this specific case.

[reds4ever]

Originally posted by KrazyHorse
The problem, of course, is in the omega-completeness...

Or induction, in this specific case.

Damn! Beat me to it!!!

[Elok]

Well, Kuci, if you let P equal the number of Polytubbies idle enough to spend time on obscure metamathematical bastard-word-games, you discover that P is an ever-increasing quantity. As such, P might be considered "infinity." As infinity is by nature not able to be described by any number of words, well, there's your answer. Now somebody turn this into a babe thread.

[duke o' york]

If C is the probability that Ming will come along and close this thread then:

post ante deletia.

[Urban Ranger]

I think the problem is with the language used.

Is the statement "Let n be the smallest positive integer than cannot be defined in fewer than twenty english words" a definition or a description of a definition?

The second problem is, of course, how do you define an integer with English words?

The third problem is, do you define an integer, or do you merely describe them?

[Sava]

dorks

[Last Conformist]

Pot. Kettle. Black.

[KrazyHorse]

Originally posted by Urban Ranger
a definition or a description of a definition

What. The ****.

[Kuciwalker]

@ KH: thanks

Originally posted by Urban Ranger
I think the problem is with the language used.

No, I think KH answered the problem correctly. There are numbers that cannot be defined in fewer than 20 words, but you can't prove it.

[reds4ever]

Originally posted by Sava
dorks

[Spiffor]

Originally posted by Kuciwalker
No, I think KH answered the problem correctly. There are numbers that cannot be defined in fewer than 20 words, but you can't prove it.

1. Write a number that cannot be defined in less than 20 words.
2. Attempt to describe it in less than 20 words and fail.
3. Let the others in the thread look for a way to describe it in less than 20 words, and fail.
4. Congrats, you have just proven that there are indeed figures that cannot be described in less than 20 words. That is, it is proven until someone shreds your claim with cleverness.

[KrazyHorse]

Do you have any idea how logic works, dude?

Just because I haven't been able to find a way to do it doesn't mean that there is no way to do it.

The Goldbach Conjecture is not proven simply because nobody's come up with an even number which cannot be written as the sum of two odds.

You cannot write down a number which you can prove to me cannot be described in less than 20 words. I guarantee it.

On the other hand, I can prove to you quite easily that there must exist some numbers which cannot be described in less than 20 words.

[KrazyHorse]

"Proven" does not equal "Not disproven yet"



It doesn't even mean "not disprovable ever". "Proven" is a positive term. To prove something you must proceed from the axioms of the formal system by the deductive rules of the formal system and arrive at the statement.

[Mercator]

Let Mercator_date^time be the number defined by me at that date and time.

Ergo, any number can be defined in less than 20 words.

QED



Stupid word games.

[civman2000]

It has nothing to do with Godel or incompleteness. It is all about inconsistency. All you've done is shown that a certain logical system--namely the form of naive set theory that allows such descriptions as "cannot be defined in fewer than twenty english words"--is inconsistent. Alternatively, you have proven that if set theory is consistent, then the predicate "cannot be defined in fewer than twenty english words" cannot be expressed in the language of set theory.

If you know some set theory, here's a similar paradox: Say that a set is definable if, well, we can define it in some way. There are clearly only countably many definable sets, since any description must be finite in length. Thus there exist some undefinable ordinals. Let alpha be the least undefinable ordinal. But we just defined it!

What this shows is that the concept of "definability" cannot itself be definable.