Getting Set

In the second half of the 19^th Century, mathematicians began looking at sets of mathematical objects  both as a  subject of study and as a tool for working in other areas.  The most influential was Georg Cantor who is commonly credited with creating set theory as a mathematical subject.  Cantor and others analyzed the characteristics not only of finite sets (such as the set of  the prime numbers less than five thousand or the set of phone numbers on the first thirty pages of the 1966 Manhattan phone book) but also of infinite sets (such as the set of all positive whole numbers or the real numbers between 8 and 11).  

It became apparent that there were interesting and important  differences between the characteristics of the two sorts of sets -  some obvious and some less so. For example if one takes the six letters, A,B,C,D,E,F and lists them in a column on the left side of a page and again in a column on the right side  and draws one line  from each object on the left to one on the right such that no two lines from letters on the left go to the same letter on the right, all six letters on the right will  have exactly one line drawn to them. It  would be impossible to do it any other way. The same is true for any finite set.  It is impossible to have a one-to-one mapping of all of a finite set A onto  only a part of A.  However with the infinite set of positive whole numbers, it is easy.  One can map each number x to two times x, getting a one-to-one function from all of the positive whole numbers to only the even ones.  If one has a finite set of mathematical objects whether or not it is defined by some rule or formula, it is possible to determine if a given object is in the set by looking at all of its members to see if any match. (Of course it the set is large enough, it may be impractical to do so, but it always can be done in principle.) If one has an infinite set of objects and no statement, rule,  or formula specifying what things are in it , it is not possible  to do this to determine whether an object is in it or not. This led some people  to wonder in what sense or even whether such sets can be said to exist.

Over time difficulties turned up within the early theory. People had assumed that given a meaningful statement about some mathematical objects,  there was a set consisting precisely of the objects for which the statement was true. Bertrand Russell showed that assumption was false, with his example of the set of all sets which are not a member of themselves. If such a set A exists, and is a member of itself, then by  the definition of A, it has to not be a member of itself. On the other hand if A is not a member of itself, by definition it is a member of itself. This and other paradoxes that came up puzzled people and made them realize that things could not be as simple or straightforward as had been assumed. 

Beginning in the first years of the last century,  a number of very talented people attempted to create a structure for a theory of sets based on axioms which, along  with rules of  logic,  would serve as a foundation for all of mathematics . Their work was driven both by a belief that such a theory would be beneficial in itself and a desire  to eliminate  or work around the  paradoxes in the earlier theory. The two best known and most accepted results are  the ZFC theory and axioms, named after Ernst Zermelo and Abraham Fraenkel, and the NGB (or NBG)  theory and axioms, named after John von Neumann, Kurt Gödel, and Paul Bernays.  The two are similar but not identical, and at various times one or the other has been more fashionable among mathematicians.  (People have developed  many  other theories  and models with postulates and results different from those of either ZFC or NGB -  including  some in which one or more of the axioms of ZFC  are false . However none of them really caught on, and they are of interest mainly to specialists in set theory and foundations. Still in a purely formal sense they are no more or less valid than the accepted ones.)   

The common opinion among mathematicians is that the attempts were successful.  ZFC or NGB provides a way to avoid Russell’s and other well-known paradoxes,  gives mathematicians useful tools, and so far has not led to any contradictory results.  

Still some people wonder about the theories, some of their results, or both.  Small but not insignificant minorities of mathematicians have denied or at least questioned the validity of the axiomatic systems and some of the results proved from them.  Some of those that don’t still may have nagging concerns. There has been  a sort of joke among mathematicians that the axiom of choice is obviously true, while the statement that every set can be well ordered is obviously false, which is awkward since the two are logically equivalent  in the accepted systems.

This all highlights a problem with axiomatic set theory, one that may be  at least part of the reason why there are so many theories and models out there. Some of the axioms are not axiomatic in the usual sense of that term once infinite sets enter the picture. The axiom of power sets, which says that if A is a set, there is a set B consisting of all the subsets of A, is obvious and safe  enough when thinking about finite sets. One can list all of them. If A has  three members, the subsets are  A itself, the three sets with two members each, the three sets with one member each, and, by convention, the empty set defined as the set with nothing in it.  Things are not so clear with infinite sets. A person could ask how  do you make a set of the subsets of an infinite set when you cannot specify what all those sets are and what each contains. That unpopular objection is not obviously wrong, and cannot be dismissed merely by pointing out that one needs the axiom to do a lot of interesting things. The controversy over the axiom of choice is well known. While it seems  obvious that  one can in principle form a set by selecting  one element from each of a finite collection of non-empty sets, some find it unobvious that one can do it when there are an infinite number of sets to select from.  In both ZFC and NGB the axiom of choice requires  that a well ordering for the real numbers exists.  No one has ever found one, and it is not absurd to wonder whether one really exists, and indeed what  “really exists”  means in this context. 

  

Both ZFC and NGB contain the axiom of infinity which says that there is a set N such that the empty set is in N and whenever x is in N so is the union of x and the set containing only x.  It is on the list of axioms because of the positive whole numbers.  However  in reality people do not believe in the positive whole numbers because of this axiom. They believe in the positive whole numbers because things exist singly and  in quantities and can be counted. People posit the axiom not for anything intrinsically obvious  or interesting in it but rather as a way to get - or rig the game to get,  depending on one’s point of view -  a representation of the already understood positive whole numbers (and eventually the rational numbers, real numbers, complex numbers, and  so on)  into the theory. One could see some backwardness here  in deciding  what should be called axiomatic, again according to the usual meaning of the word. 

In the time since Cantor,  people have done some remarkably clever and interesting things in  axiomatic  set  theory.  However it is appropriate to ask if the theory has done all that much –or even anything -  to enhance the believability or increase the level of certainty of ordinary mathematics. That is an interesting question which can lead a person to some useful thinking about what he believes in mathematics, why, and how.