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I wonder if there is some advantage in using individuals when defining set theory and if this has something to do with the use of classes. This is essentially because I have seen that some books start by defining the empty set whereas others consider it to be a primitive symbol. In the first case there are only sets in the domain of discourse, but in the second there are also what they call individuals. What is the purpose of using individuals compared to the other alternative?

Also, the way I understand individuals is that they are entities at the begining of the hierarchy, at the same level of the empty set, but maybe I'm misundersanting this. Could you guys please help me claryfy this...

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What do you mean by individuals? –  Asaf Karagila Sep 20 '13 at 5:05
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I still don't understand. In modern set theory, the objects of the universe are sets. So $x$ is a quantifier over sets, as well as $A$. –  Asaf Karagila Sep 20 '13 at 5:24
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Individuals are just the members of the domain of discourse. So if you start with a domain of sets and nothing else, sets will be considered your 'individuals'. You can then define individual constants (= nullary function symbols) and link them to particular sets. I don't understand what the 'alternative' is. –  Hunan Rostomyan Sep 20 '13 at 5:25
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@DanielaDiaz Section 2.2, I see now. Pretty confusing. I'm looking forward to a clarification of the issue. –  Hunan Rostomyan Sep 20 '13 at 6:08
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@AsafKaragila Now that I think about it, maybe you're right and hopefully the term "individual" is not used anymore in modern set theory and everything is a set as you suggest. I say this because this book of P. Suppes dates from 1960. –  Daniela Diaz Sep 20 '13 at 6:13

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up vote 8 down vote accepted

There are some interesting remarks, both conceptual and historical, about the role of individuals in set theory in Michael Potter's highly regarded Set Theory and its Philosophy (OUP 2004) -- which is a rewrite of his earlier Set Theory text book but now with more conceptual/historical commentary (hence the revised title).

Potter starts by constructing a hierarchical set theory which does have individuals (urelemente, as some say) at the bottom level. After all this is the natural initial approach: we begin with some things (numbers, space-time points, whatever) and then start considering sets of them, and then sets of what we've got so far, and so on up ... And only after having set up an axiomatic framework for a theory with individuals, Potter considers what happens if you now add an Axiom of Purity (as he puts it) to the effect that there are, after all, no urelemente -- the pure sets with no individuals in their transitive closure are the only sets. He writes:

The axiom of purity, or something equivalent to it, is assumed in almost every modern treatment of set theory in the literature. The main reason for this is that, as was discovered fairly early, it is not necessary to assume the existence of individuals in order that set theory should act as a foundation for mathematics, while if we rule them out from the outset, we can simplify the theory, getting rid of one primitive and tidying up the development considerably. If the only objective is to give mathematicians a theory which can act as a foundation, it is inevitable that they will choose the one that seems simplest to them.

Indeed individuals would probably have made an exit from set theory earlier than they did if it had not been for an accident in the progress of work in the metatheory. In 1922 Fraenkel discovered a method for showing the independence of the axiom of choice from theories such as ZU [Zeremlo set theory with urelemente] that allow individuals. This method was refined (Lindenbaum and Mostowski 1938) and then exploited by others (e.g. Mendelson 1956; Mostowski 1945) to prove the independence from such theories of various other set-theoretic claims. It was not until 1963 that Cohen showed how to convert this into a method that works for theories like Z which ban individuals. In the intervening period there was therefore a reason for set theorists (who tend, after all, to be the people who write set theory books) to regard permitting individuals as worth the extra effort. After 1963, however, not even set theorists had any use for individuals. Worse, there are proofs in set theory that do not work if we have to allow for them. So it is unsurprising that in the last 40 years individuals have largely disappeared from view.

For further related discussions about basic issues, see Potter's immensely helpful book. @NeilBarton then gives some excellent further suggestions for intriguing glimpses beyond.

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So individuals (or "urelemente" as they are sometimes called) are used very little in modern mathematics. This is partly because the set theoretic hierarchy is very big; we can represent most mathematics in the first few infinite levels, so the urelemente are simply an unnecessary complication.

However, set theory with urelemente has been put to interesting *meta*mathematical use. For instance, Vann McGee has shown that (given unrestricted first order quantification) one can prove that the pure sets in any model of second-order $ZFCU$ ($ZFCU$ is standard second-order $ZFC^2$ with two further axioms added; that there is at least one urelement and that the urelemente form a set) are isomorphic. This contrasts with the corresponding result for $ZFC^2$ that given any two models of $ZFC^2$ either the two are isomorphic or one is isomorphic to a proper initial segment of the other. Thus the addition of urelemente here allows one to pin down the pure sets of the theory up to full isomorphism; something which is not possible in $ZFC^2$.

The reference for the McGee is the following:

Vann, McGee-How We Learn Mathematical Language, The Philosophical Review, Vol. 106, No. 1 (Jan., 1997), pp. 35-68

As for the relationship between individuals and classes; I don't see a link clearly myself. However, the following is a paper in which you might be interested, in which Christopher Menzel argues that ordinals should be thought of as urelemente and considers a set theory of sorts where the ordinals exist in $V_0$ and form a set at $V_1$. In this case we have what we'd normally think of as a proper class (the ordinals) forming a set on the basis of a specific view about the nature of urelemente. The reference for that paper is:

Menzel, Christopher (1986). On the iterative explanation of the paradoxes. Philosophical Studies 49 (1):37 - 61

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