Transitive Sets and the Distinctness of Sets From Their Members

Vlastimil asked for examples of transitive sets. A transitive set is a set every element of which is a subset of it. (Hrbacek and Jech, Introduction to Set Theory, p. 50) There is no lack of examples. The null set vacuously satisfies the condition 'if x is an element of S, then x is a subset of S.' The set consisting of the null set — {{ }} — is also transitive: it has exactly one element, the null set, and that element is a subset of it because the null set is a subset of every set.

Now consider the set consisting of the foregoing two sets, the null set and the set consisting of the null set: {{ }, {{ }}}. This set has two elements and both are subsets of it. The null set is a subset of every set, and the set consisting of the null set is also a subset of it in virtue of the fact that the null set is an element of it.

If we identify 0 with the null set, and 1 with the set consisting of the null set, and 2 with the set consisting of the null set and the set consisting of the null set, then 3 will be the set whose elements are the elements of 0, 1, and 2 which is: {{ }, {{ }}, {{ }, {{ }}}}. This last set has three elements and each is a subset of it. One can continue like this and generate as many transitive sets as one likes. For each natural number there is a corresponding transitive set.

Now how does all this bear upon my assertion that a (mathematical) set is an entity 'over and above' its members (elements)? That sets are treated in set theory as single items 'over and above' their members can be seen from the fact that some sets have sets as members without having their members as members. The power set of {Socrates, Plato} has {Socrates} and {Plato} as members, but it does not have Socrates and Plato as members. Therefore, {Socrates} is distinct from Socrates, and {Plato} from Plato. For if these singletons were identical to their members, then the power set would have Socrates and Plato as members.

Vlastimil seems to think that the existence of transitive sets is somehow at odds with the claim that sets are distinct from their members. Or perhaps he thinks that some sets are distinct from their members and some are not. So consider {{ }, {{ }}}. This is a transitive set since every member of it is a subset of it, which is equivalent to saying that every member of a member of it is a member of it. Thus { } is a member of {{ }}, which is a member of {{ }, {{ }}}. But although every member of the set in question is a subset of it, this does not alter the fact that the set is distinct from its members.

So I'm not sure what Vlastimil is driving at.

Note that if every member of a set is a subset of it, this is not to say that every subset of it is a member of it. {{ }, {{ }}} has itself and {{{ }}} as subsets but not as elements. Only if there were a set all of whose members are subsets of it and all of whose subsets are members of it could one argue that there are sets for which the membership and subset relations collapse, and with it the distinction between a set and its members.

Posted

January 7, 2009

Set Theory

Bill Vallicella

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2 responses to “Transitive Sets and the Distinctness of Sets From Their Members”

Vlastimil Vohánka

January 8, 2009

Bill,
Thanks!
First, yeas, I forgot about the technique of defining/modeling numbers in terms of iterative hierarchy rising from the null set. So, yes, there is no lack of examples. Now I want to ask: is there some transitive set which does not belong to this hierarchy? And if there isn’t, then necessarily so?
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„A transitive set is a set every element of which is a subset of it. (Hrbacek and Jech, Introduction to Set Theory, p. 50).“
The defining condition is: if x is an element of S, then x is a subset of S. In other words, not-(x is an element of S and x is not a subset of S).
Now let’s have the set S1 = {Socrates}, and the set S2 = {Socrates, Plato}. Neither S1 nor S2 is a transitive set, as defined by you (and Hrbacek and Jech). Socrates is an element of S1, but not a subset of S1. Similarly for Socrates, Plato, and S2.
Now cf. Oppy’s definition (books.google.com/books?id=FPU9tzW-2HAC, pp. 23-24):
„A set x is transitive if every member of x is a subset of x, that is, every member of a member of x is a member of x.“ The first part is identical to yours. What about the second part? Oppy suggests it’s equivalent to the first part. But is it really? I understand the second part as equivalent to this defining condition: if x is an element of S and y is an element of x, then y is an element of S. In other words, not-(x is an element of S and y is an element of x and y is not an element of S). But note that both S1 and S2 satisfy this defining condition vacuously because no element (Socrates, Plato) in them has an element (in the set-theoretic, mathematical sense): neither Socrates nor Plato is a (mathematical) set. Thus, S1 and S2 are transitive sets, as defined by the second part of Oppy’s definition.
Wikipedia entry on transitive set is similar to the second part of Oppy’s definition: „In set theory, a set (or class) A is transitive, if whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently, whenever x ∈ A, and x is not an urelement, then x is a subset of A.” en.wikipedia.org/wiki/Transitive_set
Again, S1 and S2 vacuously satisfy these defining conditions because no element of them has an element, and because they have only urelements (i.e., elements which are not non-empty sets).
You said that the set {{ }, {{ }}} „… is a transitive set since every member of it is a subset of it, which is equivalent to saying that every member of a member of it is a member of it.“
Well, I accept that any set belonging to the iterative hierarchy rising from the null set is such that every member of it is a subset of it, and also that every member of a member of it is a member of it. (Though right now this is not clear to me.) Still, the set S2 (and also the set S1), which isn’t iterative, is (vacuously) such that every member of a member of it is a member of it, but it is not true that that every member of it is a subset of it.
The Wikipedia entry on transitive sets does not seem to explicitly confine its claims to iterative sets. On the other way, Oppy at least indicates such a restriction. P. 21 states: „We shall not suppose that there are any individuals.“ He means that he will not suppose that there are any urelement different from the empty set. On p. 22 we can read a stronger statement: „There are various changes that we could make to our axioms. … we could weaken Axiom 1 [i.e., if two sets have the same members, then they are identical] to allow individuals (ur-elements) other than the empty set …“ This suggests that the version of set theory which Oppy in fact presents (as opposed to the version he could present) not only does not presuppose urelements different from the empty set, but even disallows such urelements. That is, the version of set theory presented by Oppy is confined to sets with no such urelements. So, he likely avoids the tension between the first and the second part of his definition of transitive set. Still, he should have been clearer.
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You wrote: “3 will be the set whose elements are the elements of 0, 1, and 2 which is: {{ }, {{ }}, {{ }, {{ }}}}.“ You should delete „the elements of“, shouldn’t you?
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A minor comment. Suppose each natural number has a corresponding transitive set. But should we „identify“ natural numbers with their corresponding transitive sets?
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How is the fact that ALL sets are treated in set theory as single items ‘over and above’ their members seen from the fact that SOME sets have sets as members without having their members as members?

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Vlastimil Vohánka

January 8, 2009

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Transitive Sets and the Distinctness of Sets From Their Members

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