Antimail

"Minus infinity"?

I was thinking minus infinity too...

According to my number theory class -- ages ago -- infinity is NaN. It's more of an innumerable.

Minus ninety

"The least number expressible in under thirteen words" is a valid English phrase using only 8 words. Therefore if there were ever such a number this would be a contradiction.

This is a special case of Russell's paradox, and is resolvable using his theory of types.

Sorry, "expressible" should obviously say "inexpressible" up there. We can make it watertight by changing the phrase to "The least integer number inexpressible using less than 13 English words".

This rules out the case of infinities as with the examples above.

A interesting puzzle based on this:

"The least integer indescribable using less than ten words" leads to the same contradiction as with thirteen words.

But "The least integer indescribable using less than two words" does not, because there is no phrase with the same meaning under two words long.

So the question is this: what is the maximum n such that "the least integer indescribable using less than n words" does not lead to a contradiction?

[I don't know the answer!]

Hmmm, what about the following argument:

English has a finite (though incredibly large) dictionary, even counting all derivative words. So we can (in principle) enumerate all phrases of length 1, length 2, length 3, etc.

So far we're just treating these things as symbols for arbitrary objects. Only THEN do we begin to interpret them. We have to keep 'symbol space' and 'interpretation space' separate in our minds.

So say if we found an example of an "contradictory" phrase of length 5. If we went through the entire list of phrases of length 4 and couldn't find a contradictory phrase amongst them when interpreted, this surely means n=5 would be the answer. And we already know n<=10...

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