Philosophical Introduction to Set Theory

Stephen Pollard

• Publisher: Dover Publications
• Year: 2015
• ISBN:
  9780486805825
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

The primary mechanism for ideological and theoretical unification in modern mathematics, set theory forms an essential element of any comprehensive treatment of the philosophy of mathematics. This unique approach to set theory offers a technically informed discussion that covers a variety of philosophical issues. Rather than focusing on intuitionist and constructive alternatives to the Cantorian/Zermelian tradition, the author examines the two most important aspects of the current philosophy of mathematics, mathematical structuralism and mathematical applications of plural reference and plural quantification.
Clearly written and frequently cited in the mathematical literature, this book is geared toward advanced undergraduates and graduate students of mathematics with some aptitude for mathematical reasoning and prior exposure to symbolic logic. Suitable as a source of supplementary readings in a course on set theory, it also functions as a primary text in a course on the philosophy of mathematics.

VIII

ITERATIVE HIERARCHIES

1. Set Formation

In Chapter 5, we explored the view that theories are the only legitimate objects of set theoretic research. We shall now see how set theorists can be viewed as scientists who investigate the properties of structures known as “iterative hierarchies.” Very roughly, an iterative hierarchy is an array of sets regarded as the product of an iterated process of set formation which begins with the empty set and then proceeds through a well-ordered series of stages, each stage containing sets whose members appear at prior stages. There are all sorts of theories about iterative hierarchies, each of which can be thought to characterize the “real, honest-to-goodness, genuine iterative hierarchy.” But, rather than endorsing a particular view about “the” iterative hierarchy, we shall speak of iterative hierarchies (in the plural). We shall view set theorists as devotees of a family of distinct structures, each of which will be accorded the title of an iterative hierarchy.

It is common to talk about iterative hierarchies coming into existence through a human, mental process of set formation stretched out over time. Since the hierarchies under consideration turn out to have enormous and enormously complex infinitary structures, this talk could be taken literally only if one gave a very peculiar and implausible account both of the nature of time and of the extent of human mental powers.⁹⁰1 acknowledge that the image of temporal, mental set formation may be a useful picture. But it is one which I feel little inclination to take seriously.

This implausible picture is tempting because, in an iterative hierarchy, sets are ordered in such a way that the members of a set are always prior in the ordering to the set of which they are the members. So if one takes this priority to be temporal, one acquires the natural image of a set being produced only after all of its members have been produced. And this allows one to make a certain amount of sense of the particular ways sets are arranged in an iterative hierarchy. For example, the position of the empty set at the bottom (or at the top-depending on how you look at it) of every iterative hierarchy becomes fairly natural: after all, the empty set is the one set which need not be posterior to its members-for the simple reason that it has none. The very expression Iterative hierarchy’ harks back to the notion that the universe of set theory is created by first forming the empty set and then iterating the process of set formation. More precisely: At time 0, one forms all the sets whose members are sets which have already been formed; that is, since no sets have been formed, one forms the empty set:

At time 1, one again forms all the sets whose members are sets which have already been formed; that is, one again forms the empty set and, further, since the empty set has already been formed, one forms the set whose only member is the empty set:

At time 2, one yet again forms all the sets whose members are sets which have already been formed:

And so on. Thus, at time 0, one forms 2⁰ sets – that is, one forms just one set. At time 1, one forms 2¹ (=2) sets; at time 2, 2² (=4) sets; at time 3, 2⁴ (= 16) sets; at time 4, 2¹⁶ (=65,536) sets. And, at time 5, one forms 2^65,536 (=???!) sets.

Of course, one really does nothing of the sort. Even if one managed to make sense of the notion of our “forming sets,” one would be hard pressed indeed to explain how it is even possible for anyone to form 2^65,536 sets. It would be harder still to establish that anyone actually does this. And note: we have so far dealt only with what is supposed to happen at very small finite time stages. One would also have to account for the formation of truly monstrous numbers of sets at time stages much later than stage 5. And I do mean much later. Depending on what iterative hierarchy one is committed to, one might even have to account for the formation of sets at a “time” stage α –where a is a remote, complex, infinite ordinal number. So, as I already mentioned, this picture of temporal set formation can trap one in a highly questionable view not only of human mental powers, but of the structure of time as well. I shall take it for granted from now on that this picture is merely...