Is God a Mathematician_ - Mario Livio [88]
The law itself stated innocently enough that the extension of a concept F is identical to the extension of concept G if and only if F and G have the same objects under them. But the bomb was dropped on June 16, 1902, when Bertrand Russell (figure 49) wrote a letter to Frege, pointing out to him a certain paradox that showed Basic Law V to be inconsistent. As fate would have it, Russell’s letter arrived just as the second volume of Frege’s Basic Laws of Arithmetic was going to press. The shocked Frege hastened to add to the manuscript the frank admission: “A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.” To Russell himself, Frege graciously wrote: “Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic.”
Figure 49
The fact that one paradox could have such a devastating effect on an entire program aimed at creating the bedrock of mathematics may sound surprising at first, but as Harvard University logician W. V. O. Quine once noted: “More than once in history the discovery of paradox has been the occasion for major reconstruction at the foundation of thought.” Russell’s paradox provided for precisely such an occasion.
Russell’s Paradox
The person who essentially single-handedly founded the theory of sets was the German mathematician Georg Cantor. Sets, or classes, quickly proved to be so fundamental and so intertwined with logic that any attempt to build mathematics on the foundation of logic necessarily implied that one was building it on the axiomatic foundation of set theory.
A class or a set is simply a collection of objects. The objects don’t have to be related in any way. You can speak of one class containing all of the following items: the soap operas that aired in 2003, Napoleon’s white horse, and the concept of true love. The elements that belong to a certain class are called members of that class.
Most classes of objects you are likely to come up with are not members of themselves. For instance, the class of all snowflakes is not in itself a snowflake; the class of all antique watches is not an antique watch, and so on. But some classes actually are members of themselves. For example, the class of “everything that is not an antique watch” is a member of itself, since this class is definitely not an antique watch. Similarly, the class of all classes is a member of itself since obviously it is a class. How about, however, the class of “all of those classes that are not members of themselves”? Let’s call that class R. Is R a member of itself (of R) or not? Clearly R cannot belong to R, because if it did, it would violate the definition of the R membership. But if R does not belong to itself, then according to the definition it must be a member of R. Similar to the situation with the village barber, we therefore find that the class R both belongs and does not belong to R, which is a logical contradiction. This was the paradox that Russell sent to Frege. Since this antinomy undermined the entire process by which classes or sets could be determined, the blow to Frege’s program was deadly. While Frege did make some desperate attempts to remedy his axiom system, he was unsuccessful. The conclusion appeared to be disastrous—rather than being more solid than mathematics, formal logic appeared to be more vulnerable to paralyzing inconsistencies.
Around the same time that Frege was developing his logicist program, the Italian mathematician and logician Giuseppe Peano was attempting a somewhat different