A cute little paradox. This shows why mathematics chose to spend a lot of time on making sure things are well defined.

If such a number exists, then "The largest number that can be described in 14 words or less plus one" is greater. Therefore no such number exists.

This node is a result of the myriad writeups to smallest number greater than 0. We have to make a distinction between English statements and mathematical statements, otherwise we are just wasting our time.

That 12-word description doesn't define a number, though its grammatical form suggests it does. An expression "the X" intended to refer to one single thing is called a definite description (the term is Russell's). It is logically composed of several assertions.

For example, "the tallest dog in Samoa" requires someting like (i) there are dogs in Samoa, (ii) dogs can be compared for tallness, (iii) among any set of dogs the tallness comparison results in a unique tallest one.

The expression "the largest... that can be..." will have different conditions. Probably the easiest way for it to fail is to attack the "can".

If English were frozen we could say there were (say) 500 000 words in the language, therefore 500 000 one-word expressions, 250 thousand million two-word expressions, and so on up to fourteen-word, and then inspect them all to see which define numbers; and of these we could determine the largest.

But in rejecting all the vast number that do not define numbers, we would reject such things as "the right cross swivel expansion of forty-seven factorial", which in the year 2270 could well be meaningful and define some newly-invented huge number.

This is known as Berry's Paradox. (Thanks to JerboaKolinowski for pointing this out.) Or the Richard Paradox, I see, that's French ree-shar.

       One       two
   Three  four five  six
 Seven      eight      nine
    ten     eleven  twelve 
      thirteen  Fourteen

The fundamental issue with saying "The largest number that can be described in 14 words or less plus one" is that it does not define a number, in the pure mathematical sense. In other words, it does not answer the question "What is the number?"

I would go with something such as :

"The factorial of Googolplex raised to the power of Googolplex, repeated ninety Googolplex times"

I leave it as an interesting mathematical exercise to show if this should preferably be defined as "the factorial of (Googolplex raised to the power of Googolplex, repeated ninety Googolplex times)", or "(The factorial of a Googolplex) raised to the power of Googolplex, repeated ninety Googolplex times". Either way, we are gazing in awe at a very large number.

We should not use constructs such as Graham's Number, as that requires a paragraph or more to define how it is constructed, and you cannot write down the number on a piece of paper to show anyone what it is.

 

 

 

 

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