# Reality Conditions

## Saturday, February 03, 2007

### Who can name the largest number?

On Friday, Jan. 26, two philosophers, MIT Associate Professor Agustin Rayo (The Mexican Multiplier) and Princeton Associate Professor Adam N. Elga (Dr. Evil) engaged in the Large Number Duel, in which they attempted to one-up each other by inscribing the largest finite number ever to be written on an ordinary-sized chalkboard. The feat, if successfully accomplished, would be worthy of a note in the Guinness Book of World Records. (...)

The rules of the duel gave free rein to the contestants’ creativity and humor, maintaining only a ban on the use of infinity, and restricting statements about the number proposed to a primitive semantic vocabulary. The battle itself was intense and the room in the Dreyfoos wing of the Stata Center was packed, with people standing on chairs and at least 20 students craning their necks from the doorway.

The contest opened in the style of a boxing match, with competitors presented “in the red corner” and “in the blue corner.” Elga went first, writing the number one. “Ha!” announced Rayo, as he countered with a string of ones across the board. Elga retaliated with a clever trick, erasing a line through the base of half of the ones to turn them into factorials. (...)

Near the end of the duel, Rayo furiously scribbled on the whiteboard: “The smallest number bigger than any number that can be named by an expression in the language of first order set-theory with less than a googol (10100) symbols.”

Although this definition took a bit of tweaking, including what Rayo described as his “second order logic trick,” it soon won him the duel.

As Elga collapsed, slain, the referee closed the ceremony.

• +1

;)

By  fh, at 11:52 PM, February 03, 2007

• How about: The smallest number bigger than any number ever to be described on a blackboard by Prof Rayo?

By  fh, at 11:54 PM, February 03, 2007

• I suspect the clause "restricting statements about the number proposed to a primitive semantic vocabulary" is intended to rule out tricks as the one second one you suggest.

But I confess I do not understand why the last one to speak could not always repeat the previous number written by the other one, adding 1. Perhaps there was a sort of "gentlemans' agreement" that every new number proposed had to use an "original" kind of definition, instead of relying on the one used by the other.

• Actually upon reflection, I think the above idea hinges on the details of the interpretation of the rules:

The trick is this: once your opponent writes the same definition on the board then the only solution is "infinity", therefore your competitor wrote infinity on the board which is forbidden.

Alternatively you could argue that there is no finite number fitting the designated criteria if both statements are on the board. Therefore neither has really written down a number then the argument fails.

By  fh, at 10:13 PM, February 06, 2007

• Eliezer Yudkowsky gives an interesting account of the largest number to be used in a serious mathematical proof, here:

http://yudkowsky.net/singularity.html#beyond_big

Sure, he's talking about the singularity, but it's mindboggling (for some reason) how big some numbers can get!

By  Jeebus, at 6:03 PM, February 15, 2007

• The largest finite real number can be defined in a sentence thus:

"The sum of all real numbers in the set of real numbers"

There cannot be a larger finite value than that, as you would have to sum a [real] number outside of the set of real numbers....a contradiction in itself.

Enjoy =)

By  Anonymous, at 6:19 PM, May 05, 2010

• Anonymous on May 5 is wrong. The sum of all of the real numbers in the set of real numbers would itself have to be in the set of real numbers. The set of real numbers is already uncountably infinite. Adding any real number to the set would actually not change the size of the set. Adding infinity to infinity would not change the size of infinity, either. I know this seems counterintuitive, but I refer you to Hilbert's "infinite hotel" paradox.

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By  Christopher, at 11:21 PM, November 25, 2011

• In theory, there isn’t an upper limit – any number you reach could just have 1 added to it to make a bigger number.

The largest number for which we have a name is “googolplex” – which is a one followed by a googol zeroes (a “googol” being a one followed by a hundred zeroes).

Practically, if you were to start counting from 1 the second you were born and increase by a number a second until you die at the age of 79 you’d reach only as high as 2,492,985,600 – less than half the number of people alive on the planet.

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• Graham's number is this big

3^^^^^...^^^^^3
3^^^^^...^^^^^3
3^^^^...^^^^^3
...
3^^^^3
64 layers

the ^ mark means un up arrow
3^3=3 to the power of 3
3^^3=3^(3^3)