The greatest number... is equal to 1 -----------------------------------
Dizzying infinity: suppose there exists a strictly positive integer N that is the greatest natural number. Then N is equal to 1. Indeed, the square of every nonzero number is greater than that number, except for the square of 1, which is equal to 1. Thus N2 must be greater than or equal to N. But since N is, by definition, the greatest integer, its square cannot be greater than N! It must therefore be equal to N: N2 = N, so N = 1.
This paradoxical proof shows that we must first establish that a problem has a solution before trying to find it (here, that there is a greatest natural number... which is false). Otherwise, we reason as we might about Roland's mare (from Orlando Furioso by Ludovico Ariosto, known simply as Ariosto): she has every virtue... except that of existing!
Just as there is no greatest integer, neither is there a set with more elements than every other set (whose cardinality would be greater than that of any other set), nor a set containing all sets. So much for Saint Anselm's argument for the existence of God: "It is evident that there exists a being than which no greater can be conceived." Along the same lines, how are we to assess God's wealth (or age...), which must be as great as possible?
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Numbers that do not... really exist! ------------------------------------------
There is no end to conjectures about numbers: "One of the most remarkable conjectures about Mersenne numbers is that the number of conjectures about Mersenne numbers will always exceed the number of known Mersenne numbers", said the mathematician Waclaw Sierpinski. Following David Hilbert, let's consider a very special number: let n be the smallest integer that will not have been mentioned during the 21st century. Now, following Max Black, let's define another number m: let m be the smallest integer that is not defined anywhere in this special issue. These two numbers, n and m, are perfectly well defined, yet would no longer satisfy their definitions if we stated them explicitly!
In 1905, when the foundations of mathematics seemed to be shaken by the paradoxes of set-theoretic logic, Jules Richard presented a paradox known as the least-integer paradox. Martin Gardner discusses it under its English name, the Berry paradox. Here is another version: the number ninety-seven thousand two hundred and ninety-seven is curious in that it can be defined as a "number not specifiable in fewer than ten words." Yet this definition itself contains strictly fewer than ten words and therefore specifies a number that should be specified only by a definition containing ten words or more...
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Give Epimenides his due... --------------------
Set-theoretic paradoxes are often regarded as paradoxes of language; their prototype is the paradox of Epimenides. "I am lying" means "I am always lying", and hence also "all the sentences I utter are false." Yet this sentence is itself an element of that set. Whoever utters it is lying, and the paradox begins with the conflation of a set (the sentences) with one of its elements (the sentence "I am lying").
According to Ludwig Wittgenstein, every language has a structure about which nothing can be said within that language itself; but there is another language whose object is the structure of the first. Carnap and Tarski clarified these levels: at the first level, language deals with objects; second-level language (or metalanguage) in turn deals with language, and so on. Consider again the sentence "This sentence is false": the first level is simply what the sentence says, and the second is what the sentence says about itself. But the sentence cannot be both object and subject, just as something cannot simultaneously be an element and a set.
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Turning defeat into victory --------------------------------
Gödel studied decidable and undecidable statements; the completeness, incompleteness, consistency, and inconsistency of theories; the possible reducibility of one theory to another; and "metamathematics." Russell and Whitehead developed the theory of logical types, which later had an unexpected impact on... psychiatry, inspiring the work of Gregory Bateson and the Palo Alto School, built around paradoxical injunctions such as "Disobey me!" or "Be natural!" All this shows the great heuristic value of paradoxes.
By using the paradoxes of set-theoretic logic as springboards to new knowledge, logicians and mathematicians ultimately turned defeat into victory. André Weil wrote: "God exists because mathematics is consistent, and the devil exists because we cannot prove it."