Set

A set is a complex whose existence depends upon all its parts existing but does not depend upon its parts existing in the right way. That suffices for a definition of ‘set’.

For whatever reason, much ink has been spilled (and computers typed) in an effort to characterize and understand sets. Some mathematicians seem to even think that all of math is ultimately a study of sets. I believe that the degree of devotion to understanding sets by contemporary thinkers is out of balance. There are other categories, some more fundamental than Set, to investigate. Why so much devotion to the kind, Set, and so little devotion to the kind, Substance?

I also think it’s a mistake to think that just because talk of some other categories, like shapes and quantities, can be expressed syntactically in terms of sets, that shapes and quantities just are sets. I might be able to represent anything using the language of sets, even my aunt Linda. But that doesn’t mean that my aunt Linda is a set. I think we can grasp things like, shapes and quantities, directly and can see (be aware of) certain relationships between certain shapes and certain quantities. By our awareness of these abstract arrangements, we can also see that shapes are not the same as quantities, and that at least one quantity—viz., oneness—is not itself a set.

Of course, it might be useful to represent things of one type in terms of things of another type. For example, I think that progress in probability theory can be aided by representing probabilities in terms of ven diagrams (shapes). But that isn’t to say that probabilities just are ven diagrams.

Here’s the ironic thing. With all the work done in investigating sets, I suspect that some of widely accepted axioms that have been given to describe sets are false. Of course, naïve set theory is false because it entails a contradiction: it entails that the set of all sets that are not members of themselves is itself a member of itself if and only if it is not a member of itself. But I am also skeptical of some of the widely accepted axioms of ZFC. For example, ZFC either implicitly or explicitly entails that there is an empty set, a thing having no members, and the axiom of extensionality entails that there cannot be more than one empty set. Well, I have no grasp of the relation is a member of if that relation isn’t a species of is a part of. And it seems to me obvious that there are many things that lack parts. They are what I call simples. So I reject the idea that there is just one “empty set.” This problem might remedied by arbitrarily designating a certain thing, maybe Being itself, to play the role of the empty set. But still, strictly speaking, it is false that there is exactly one thing that lacks members (or parts).

Now someone might reply that ZFC set theory isn’t trying to describe what I call, ‘set’. Indeed, someone might view the axioms of ZFC as a kind of implicit definition of the term, ‘set’. Fair enough. But then I confess that I have no idea at all what ‘is a member of’ is supposed to mean. I do grasp ‘is a part of’ because I grasp immediately a certain mereological relation holding between the top-half of my visual field, say, and my complete visual field. So if ‘is a member of’ isn’t that relation, then I really don’t know what relation is being expressed. I believe that all the theoretical work that can be done using ZFC sets can be done using Rasmussen sets (sets according to my definition of ‘set’), and Rasmussen sets, unlike ZFC sets, are things that I comprehend.

One virtue of my definition of ‘set’ is that it’s a simple definition that leaves it wide open what their existence and identity conditions are. Also, most of the axioms of ZFC probably are true of Rasmussen sets. It’s just that we need to do some tweaking. We can accept the following as is:

(1) Axiom of extensionality: Two sets are identical if and only if they have the same parts.

(2) Axiom schema of specification: If z is a set, and $\phi\!$ is any property that certain of the parts of z each exemplify, then there is a set y whose parts are just those parts in z that exemplify the property.

(3) Axiom of regularity: Every set x has a part y such that x and y do not share any parts in common.

But consider this one:

(4) Axiom of pairing: If x and y are sets, then there exists a set which contains x and y as parts.

I have an analogous axiom for arrangements. But I could see someone being skeptical here as applied to sets. For example, one might think it is redundant to posit a thing that has A, B, and C as parts in addition to a thing has these five parts: A+B, B+C, A, B, and C. Now it turns out that the negation of (4) can be derived from my more complicated definition of whole if we include an axiom that says that parthood is a transitive relation. But I’m open to someone denying the transitivity of parthood (just as is a member of isn’t thought to be transitive). Also, I’m really only committed to the simpler definition that a complex is something that has at least one part. At any rate, I think theorists need to think more carefully about (4). Let us not just assume it is true. Let’s try to find out if it’s true, if we can.

Consider next:

(5) Axiom of union: For any set $\mathcal{F}$ there is a set A containing every set that is a part of some part of $\mathcal{F}.$

But keep in mind that someone might question whether sets can themselves be parts (members) of sets.

(6) Axiom schema of collection: Let $\phi \!$ be any formula in the language of ZFC whose free variables are among $x,y,A,w_1,\ldots,w_n \!$ . So B is not free in $\phi \!$ . $\exists ! y$ is a quantifier binding y, meaning that exactly one $y\!$ exists, up to equality. Then: $\forall A\,\forall w_1,\ldots,w_n [ ( \forall x \in A \exists ! y \phi ) \Rightarrow \exists B \forall x \in A \exists y \in B \phi].$

I have only a vague sense of what this axiom is supposed to say. Part of the problem is that it contains free variables. And I’m skeptical that any expression containing a free variable actually expresses a proposition. Someday, I may try to translate this into what I take to be an intelligible sentence. For now, I leave the matter open.

Turn now to:

(7) Axiom of infinity: Let $S(x)\!$ abbreviate $x \cup \{x\} \!$ , where $x \!$ is some set. Then there exists a set X such that Being is a part of X and for any set y that is a part of X, $S(y)\!$ is also a part of X.

This entails a thing with infinitely many parts. But not only that, it entails that all its parts are ultimately built out of just Being. For example,

X = {Being, {Being}, {{Being}, Being}, {Being, {Being}, {{Being}, Being}}…}.

Is there really a thing like that? I suppose that will depend in part on whether you think that sets can themselves be parts of sets. Note: some mathematicians seem to think that the members of X just are the natural numbers, where Being is replaced with the entity, 0 or 1.

Here’s the final axiom of ZF:

(8) Axiom of power set: Let $z \subseteq x$ abbreviate $\forall q (q \in z \Rightarrow q \in x).$ For any set x, there is a set y which is a superset of the power set of x.

That seems reasonable to me if sets can themselves be parts of sets. Now for the notorious axiom of choice (the ‘C’ in ZFC):

(9) Let X be a set whose parts are all complexes. Then there exists a function f, called a "choice function," whose domain is X, and whose range is a set, called the "choice set," each member of which is a single member of each member of X.

If parthood is transitive, then I very much doubt that anything has only complex things as parts. And in that case (9) is trivially true. But we can avoid that worry by simply letting X be any arbitrary set that contains some complexes as parts. The question that mathematicians debate is whether this is always true when X is infinite. I’m inclined to think that this debate is misguided. Those who are skeptical are skeptical because they think that every set should be “constructable” in principle. I suspect this thinking may be residue from the days of logical positivism. I wonder: why think that one must be able to represent something by means of a finite series of steps for it to exist. Indeed, on my understanding of propositions, there are far more propositions than there obtainable types of finite linguistic representations (because there is a distinct proposition about each and every type of finite linguistic representation, plus propositions about all the subsets of the set of all such types). Thus, many propositions cannot be expressed by a finite linguistic representation. If this isn’t a problem for propositions (ordered complexes of individual essences), then I don’t see why there should be a special problem here for sets (unordered complexes).

Metaphysicians need not stop with just these axioms. We can consider others, too. For example, we may wonder what kinds of things can be parts of a set. Can anything from any category be part of a set? Or is there a restriction here. Perhaps only abstract things can be parts of sets. Many people find it odd to think there is a thing that contains just my left toe plus the moon, for example. Maybe, then, sets cannot contain material parts. However, for a theory in which the points of space and bits of matter are the primary parts of all sets, see Sets @ TWOW. This question and others still require further investigation. I think it is a serious mistake to view these questions as somehow out of bounds just by definition. (Or if they are out of bounds by definition, then these questions can simply be dressed up in different words!)

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