The List of All True Statements

Earlier we noted that it is not possible to have a correct and an extremely detailed prediction of the future, if the prediction is known to people.

In a similar way, there are often restrictions on the possibility of bringing together all true statements of a certain form. This is frequently independent of whether people know of them or not.

For example, consider the library of Babel. A great deal is contained in it, including every sentence in this blog post, as you can discover by testing the site’s search function. Suppose someone were to go through these texts and make a list of all the true ones, avoiding meaningless, indeterminate, and false ones. In time, he would come upon this page, which has the sentence, “this sentence is not contained in your list of true sentences.” Should he include this sentence, or not?

If he does not include the sentence, he will have missed a true sentence. And if he does include it, he will have included a false one. So it is impossible in principle for his project to succeed: he cannot make a list of all the true sentences in the library.

This is related to the Paradox of the Liar, but it is not the same thing. Whether or not a sentence is contained in someone’s list is a perfectly objective fact. It is either there, or not, and there is no inconsistency in reality on this account. But the list maker still cannot succeed. In practice this is a case of the Liar Game.

 

 

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