What is "The smallest positive integer not describable in less that twelve words"?

Question

you make a very good parrot profit0004. im sure widipedia is happy. do you have an answer to the question, or not?

almost a good follow-up question MindscapeWanderer, but there is no smallest integer....

Answer 1

Doesn't exist.

Just because a property CAN apply to some number, doesn't mean that in necessarily does.

The smallest positive integer not describable in less that twelve words?
What is the last integer?
What two integers do you take the quotient of to find Pi?

They are statements about integers that need not apply to any of them, and we know they don't because they result in a contradiction if they did.

Here is one that does exist

What is the smallest positive integer you cannot name?

It has an answer, but you'll never tell me.

Have fun.

Point of Clarification: The previous answer, though incredibly witty, is incorrect because that number CAN be named in less than 12 words.

In response to your response; are you sure, and why are you sure? I ask this because I AM sure that it exists. The reason why is a very interesting road if you want to go down it.

edit: I see your beef with my question now and I fixed it. I wouldn't say very large negative numbers are "small" if that's what you were going after. They are large numbers that are negative. Linking why to the real world, say you are $100,000 dollars in debt. Is your net worth of -$100,000 dollars a small integer; smaller than if you were at -$1?
In all fairness though I should have been more specific.

Answer 2

The Berry paradox is the apparent contradiction that arises from expressions such as the following:

The smallest positive integer not nameable in under eleven words.

For the sake of simplicity, we assume here that hyphenated words like "self-evident" do not count as a single word but as several words ("self" and "evident").

We can argue that this phrase specifies a unique integer as follows: there are finitely many phrases of fewer than eleven words. Some of these phrases denote a unique integer: For example, "one hundred thirty six", "the smallest prime number greater than five hundred million" or "ninety raised to the centillionth power". On the other hand, some of these phrases denote things which are not integers: For example "Miguel de Cervantes" or "Mount Kilimanjaro."In any case, the set A of integers that can be uniquely specified in under eleven words is finite. Since A is finite, not every positive integer can be in A. Thus by well-ordering of the integers, there is a smallest positive integer that is not in A.

But the Berry expression itself is a specification for that number in only ten words!

This is clearly paradoxical, and would seem to suggest that "nameable in under eleven words" may not be well-defined. However, using programs or proofs of bounded lengths, it is possible to construct an analogue of the Berry expression in a formal mathematical language, as has been done by Gregory Chaitin. Though the formal analogue does not lead to a logical contradiction, it does prove certain impossibility results, including an incompleteness theorem similar in spirit to Gödel's incompleteness theorem; see Kolmogorov complexity for details.

The Berry paradox was proposed by Bertrand Russell (Russell, 1906). He attributed it to G. G. Berry of the Bodleian library (c.f. Russell and Whitehead 1910), who had suggested the idea of considering the paradox associated to the expression "the first undefinable ordinal".

Answer 3

The question is ill posed because the statement is inconsistent with itself.

Proof by contradiction:
Assume that the number exists.
If this number exists, then it is described using the 11 words, "The smallest positive integer not describable in less than twelve words."
Therefore this number is describable in less than 12 words.

Contradiction: therefore this number cannot exist.

For more fun look into the Halting Problem and into Godel's Incompleteness Theorem.

Answer 4

It is "The smallest positive integer not describable in less than twelve words + zero."

Answer 5

Twenty one million one hundred twenty one thousand one hundred twenty one. Perhaps?