# comprehension and ideal elements

A not uncommon thought in philosophy is that we should distinguish (in philosophy, anyway) between "sparse" ("real", "serious") and "abundant" ("ideal", "superficial") properties/classes and relations. The abundant ones are the things we get from axioms -- the comprehension scheme in the context of type theory, and set existence principles in the case of set theory. But if you are willing to countenance ideal classes (or relations or whatever), it might seem like just a prejudice to not also countenance ideal elements. Moreover, the history of mathematics is filled with examples of cases where we expanded our ontology by allowing for specific ideal objects like the square root of minus one. (Also, though its connection to my question is hardly clear, forcing is often seen as a means of adding ideal elements.) My general question is: what kinds of systems have been studied that include the idea of postulating ideal elements and/or allow for greater symmetry between the treatment of individuals and classes?

My first impression, having thought about the matter not too long, is that combining comprehension principles with principles allowing the postulation of ideal elements tends to lead to inconsistency. For instance, suppose we just have a two-sorted system, with individuals and sets of individuals, and even suppose that comprehension is restricted in the way it is in ZFC. So we have the scheme

(1) There is a set such that, for every individual $s$, $s$ is in it iff $s$ is in $X$ and $\phi(s)$.

(2) There is an individual such that, for every set $Y$, $Y$ contains it iff $Y$ contains $t$ and $\phi(Y)$.

Together they lead to a contradiction if we assume that some individual $t$ is in two sets, for by (1) we can form singleton $t$, and by (2) we can find a member of singleton $t$ that is in no set but singleton $t$. Here are three things one might be able to do. First, one might be able to symmetrically restrict both comprehension and inverse comprehension in some way that would result in them not being inconsistent with one another. Second, one might try to build a system using some powerful form of inverse comprehension but without any ordinary comprehension or with only some very weak form of ordinary comprehension. If that could be done, it would show that, although we cannot unrestrictedly have both ideal classes and ideal elements, the choice of which to have is somehow arbitrary. Third, one might might try to build up a model iteratively by alternately applying (forms of) comprehension and inverse comprehension to what one has gotten so far. Have any of these avenues been pursued? Also, are there any natural ways of "inverting" any of the other set existence principles used in set theory?

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Your inverse comprehension principle (2) contradicts the pairing axiom, even in the weak form asserting only that for any two objects, there is a set containing both of them as elements. Let $\phi(Y)$ be any contradictory property, such $Y\neq Y$, which never holds of any object $Y$. In this case, principle (2) is asserting that for every $t$, there is an object $c$ such that no set $Y$ containing $t$ also has $c$ as an element. This contradicts the weak pairing axiom, which asserts that there is a set having $t$ and $c$ as elements.
Meanwhile, of course one can rescue the inverse idea, simply by stating all the axioms of set theory in inverse form. The resulting theory would be equiconsistent with the orginal theory, since it amounts just to replacing all occurences of $\in$ in the axioms with $\ni$, a mere cosmetic syntactic change. But I think you want to consider the inverse comprehension principle with a real set theory, which includes the most basic principles of set theory, and I would take this to include the weak pairing axiom. So the fact that inverse comprehension appears to contradict weak pairing may be bad news...
Update. In a positive line, let me point out that the standard ultrapower construction is essentially a method to add "ideal elements" in the sense of your question. Suppose that $M$ is a model of set theory, and that $U$ is an $M$-ultrafilter on a set $I$. The ultrapower $M^I/U$ has a copy of $M$ inside it (the range of the map $x\mapsto [c_x]_U$, where $c_x$ is the contant $x$ function), and so it may be viewed as an extension of $M$ with new elements. Thus, let us regard $M$ as a submodel of $M^\ast=M^I/U$. The ideal element $i=[id]_U$ has the property that a set $A\subset I$ is in $U$ if and only if $i\in A$ in the ultrapower. In particular, as far as subsets of $I$ are concerned, $i$ has a property $\varphi(i)$ in $M^\ast$ if and only if $U$-almost every $a\in I$ has $\varphi(a)$ in $M$. So $i$ has all and only the properties that hold almost everywhere.
This perspective is essentially equivalent to ultrafilters, since if you have $M\prec M^\ast$ and a set $I\in M$ gains a new element $i\in I$ in $M^\ast$, then you can define the ultrafilter $U$ by $A\in U\iff A\in M$ and $i\in A$. It follows that $\varphi(i)$ holds in $M^\ast$ if and only if $\varphi(a)$ holds in $M$ for $U$-almost every $a$.
I take it as devastating, if one wishes to regard the theory as a theory of sets, since a violation of pairing is so fundamental. (But I also take a violation of such weak forms of comprehension as in your question also as a fundamental objection.) But if one wants merely to develop a system around inverse comprehension, as an axiomatic system of a directed graph of a certain kind, then my second paragraph indicates a way to do it---just view $\in$ as $\ni$ in all the ZFC axioms. This amounts to a perspective by which one views the containing set as an "element" of its members. –  Joel David Hamkins May 15 '12 at 18:07