mathematical paradox?

[jinydu]

Here's a paradox that's not so easily dismissed:

Statement: All positive integers can be completely and unambiguously described in 17 English words or less.

Proof:

Assume there were positive integers that could not be completely and unambiguously described in 17 English words or less.

Then, by the well-ordering principle, there exists a smallest positive integer that means this criteria.

But this number can be described as "The smallest positive integer that cannot be completely and unambiguously described using 17 English words or less".

Note that that description uses 17 English words.

That is a contradiction. Therefore, there do not exist any positive integers that cannot be completely and unambiguously described using 17 English words or less.

[geoff]

Originally Posted by jinydu

Assume there were positive integers that could not be completely and unambiguously described in 17 English words or less.

Then, by the well-ordering principle, there exists a smallest positive integer that means this criteria.

The well-ordering principle is a set axiom, so you are assuming that "positive integers that could not be completely and unambiguously described in 17 English words or less" is a set. The resulting contradiction then proves that it is not a set.

[xilman]

Originally Posted by geoff

The well-ordering principle is a set axiom, so you are assuming that "positive integers that could not be completely and unambiguously described in 17 English words or less" is a set. The resulting contradiction then proves that it is not a set.

Alternatively, one could consider the form/meaning dichotomy.

Consider all permutations of 17 or fewer English words. This is a finite set and so each permutation, if used to label precisely one integer, labels a finite set of integers. Precisely one of those integers will be labeled with that particular permutation.

If one takes the sequence of words purely as an arbitrary label and does not attempt to interpret them otherwise, there is no contradiction and no paradox.

Paul

[JuanTutors]

Random comment: If you type in the character "." and then let a monkey press the keys on a keyboard with just the numbers 0 through 9 showing, and let him do it for ever and ever, that number will approach a transcendental number in the sense that, whatever nontranscendental number you pick, the monkey will type a number that will, in finite time, be different from yours.

So , if you want a transcendental number that doesn't have all patterns possible, just cover up the 7 key and start reading what the monkey types. And get a fast-typing monkey.

[Numbers]

Originally Posted by cheesehead

It gets better as it goes along, like many of our threads.

Cheesehead, I believe that there should be some sort of an award for getting Britney Spears, Hedy Lamarr, Semiconductor Physics, Edge Emitting Lasers and spread-spectrum radio transmission into one post in a thread that also contains reference to fast typing monkeys and a link to a site that discusses whether or not beer is cheaper in the square root of 2. But off the top of my head I can't think what the award might be called. Suggestions anyone ?

[xilman]

Originally Posted by Numbers

Cheesehead, I believe that there should be some sort of an award for getting Britney Spears, Hedy Lamarr, Semiconductor Physics, Edge Emitting Lasers and spread-spectrum radio transmission into one post in a thread that also contains reference to fast typing monkeys and a link to a site that discusses whether or not beer is cheaper in the square root of 2. But off the top of my head I can't think what the award might be called. Suggestions anyone ?

He should be granted the title "Lord High Everything Else".

Paul

[cheesehead]

Originally Posted by xilman

He should be granted the title "Lord High Everything Else".

No, no, the award should go to the thread, not to any individual.

Besides, did you actually read "Britney's Guide to Semiconductor Physics" or Hedy's patent?

[xilman]

Originally Posted by cheesehead

No, no, the award should go to the thread, not to any individual.

Fairy nuff.

Originally Posted by cheesehead

Besides, did you actually read "Britney's Guide to Semiconductor Physics" or Hedy's patent?

What? Me, read the references? You can not be serious!


Paul

[Rockfang]

Originally Posted by cheesehead

It gets better as it goes along, like many of our threads. Please read at least down to the Britney Spears photo. And the second page is better (not counting photo).

( Why Britney Spears? Well, it may be because of the "Britney Spears guide to Semiconductor Physics: semiconductor physics, Edge Emitting Lasers and VCSELs" page at http://britneyspears.ac/lasers.htm Or not.

But be sure to read how actress Hedy Lamarr (a genuine electrical engineer) co-invented spread-spectrum radio transmission after she escaped to the U.S. from pre-WWII Austria. U.S. Patent 2,292,387 for the "Secret Communication System" was granted on August 11, 1942. The patent is actually under her married name at the time - Hedy Kiesler Markey.

http://britneyspears.ac/physics/intro/hedy.htm)

Sorta OT, but wasn't there a Hedy Lamarr in the movie Blazing Saddles? ;)

[Numbers]

Originally Posted by Rockfang

... but wasn't there a Hedy Lamarr in the movie Blazing Saddles? ;)

It would seem not. There was a character in the movie called Hedley Lamarr, played by Harvey Korman.

http://www.imdb.com/title/tt0071230/

[Rockfang]

Hehe. I knew I was pretty close. :)