Thread: How does Zermelo Frankel theory avoid Urelements?

According to wikipedia, urelements (objects that are elements of a set but which are not sets themselves) do not arise in Zermelo Frankel set theory. The following operations in ZF seem to me to generate urelements though.

Let Inf be a set with a countably infinite number of elements. The existence of such a set is entailed by the Empty Set Axiom and the Axiom of Infinity.
Let Re be the powerset of Inf. The Power Set Axiom entails that Re is a set, because Inf is.

But what about the elements of Re? Any element in Re that can be described by a finite formula in the language of ZF is a set, because we can use that formula in the Axiom Schema of Separation (aka Specification) to isolate that element and assert that it is a set. But only countably many elements can be so isolated and given set status. We know that the cardinality of Re is the same as that of the reals, so there will be uncountably many other elements of Re for which there is no way they can be 'turned into sets' via the ZF axioms. Or at least, I can't currently see a way to do so.

If there is indeed no way to do this then all those other elements of Re are not sets, and hence are urelements, contrary to the claim in wikipedia.

Is there a way to resolve this difficulty?

Thanks in anticipation.

the difficulty you point out is the constructivists' main bone of contention with ZF set theory. the problem lies with the power set axiom, if S is an infinite set, it is impossible to explicitly construct ALL the given subsets.

in a naive fashion one can say: "most real numbers are totally unknowable" (for to be described in any fashion, we have to give SOME adequate description, and surely there are only countably many of these, since we can only produce countably many "specification" schema in the finite time that is human history).

what happens is you get a version of downward lowenheim-skolem: a countable model of the real numbers (the ones we can actually define). it has been suggested by some that the power set axiom is "too wild", it allows for things to be automatically sets that we have no way of actually verifying WILL be sets (in an intuistic sense).

put another way: it is easier to decide logical questions in a first-order theory. but first-order theories have limitations: one of them is that "countability" is not absolute, but depends on which model you are using. because of this, most mathematics is actually done in second-order logic (but can be done in first-order logic with a few mental gyrations).

in actual point of fact, ZF set theory was originally formulated using urelements, but it turns out that "you don't get anything extra by including them". for example, there is little point in separating the "definable" real numbers from the "undefinable" real numbers, when constructing a proof of the mean-value theorem.

in other words: it is not necessary to have an EXPLICIT formula for defining a set, in order to know that something IS a set. we just need "other sets we can make A from" in order to say A is a set. i like to paraphrase this as:

"we have no freaking idea what sets are".

because of this, i myself prefer a more structuralist (categorical) view: it doesn't matter what sets ARE, as long as they behave themselves. i don't believe, in general, that we can form a "universal foundation" for mathematics, or even that such a thing is desirable: but rather, we should tailor our approach to the needs of the problems we are trying to solve.

in a more pointed answer to your question: the subsets of Re are sets, by DEFINITION (it follows from the axioms of ZF). the fact that we don't know "which" sets all of these are, is immaterial. if ask you to give me a SPECIFIC example of a urelement, as soon as you produce it: oops, now its not anymore.

let me be a bit more specific: assume we use the "cauchy-sequence of rational numbers" definition of the reals. the fact that i cannot list ALL the possible "convergence values" of all cauchy sequences, does not stop me from using their PROPERTIES (as limits of cauchy sequences) to deduce facts about them, as these properties can be "squeezed" from rational numbers, which no one has any quibble with (the one property of the reals that is troublesome, from a logical standpoint, is the "least-upper-bound property" (every non-empty set A has a supremum)). in most proofs that use the least-upper-bound property, there isn't any need to know what a real number x actually is (that is which cauchy sequence(s) of rationals its the limit of). most theorems in analysis (or calculus, in particular) are "non-constructive", we assert the existence of real numbers (such as the "c" in the mean-value theorem) without actually producing them. i find it preferable to assume we are working in a complete ordered archimedean field, without worrying overmuch about how we obtained it. realistically: how important is the "existence" of the real numbers? it's an IDEA. if we can communicate this idea effectively and consistently, that matters more than "what is its "true nature" ".

Thank you Deveno. I like your comment that we have no idea what sets are. It seems to be true, and is a very surprising realisation to me. I came into this question by route of Skolem's 'paradox' and although I find Skolem's resolution of that perfectly convincing, this question here, which it has led to, fascinates me.

it shouldn't be THAT surprising. for example, we have no idea in geometry what lines are, or points. but that doesn't diminish the usefulness of these ideas. if anything, it increases their usefulness, by allowing us flexibility in how we apply them to the world around us. understanding is a process of drawing correspondences between disparate things. these correspondences need not be perfect nor complete to help us.

Although we introduce points, lines and planes to geometry as featureless objects, they acquire features via the axioms we place around them, eg Hilbert's 20-21 axioms. A key consequence of those axioms is that we can construct tests for whether an object is a point, plane or line. For example a set of points p is a plane iff:

1. There exist three points in p that do not lie on the same straight line; and
2. For any four points a,b,c,d in p, every pair of straight lines through two of the points (ie pairs taken from {ab, ac, ad, bc, bd, cd}) has a non-null intersection.

There doesn't seem to be any similar test for whether an object is a set. The ZF axioms provide a means for constructing sets, so that we can be sure an object is a set if it can be constructed from the axioms. But there are many more objects that cannot be so constructed, and ZF doesn't seem to provide us with any means for determining whether any one of them is a set or not.