Unrestricted Comprehension of Sets

This post deals with answering the question about what an Unrestricted Comprehension amounts to as it involves sets.

Sets are collections of things. An Unrestricted Comprehension will naturally be unrestricted about what sorts of things are to be capable of being included in sets. Thus, any specifiable thing at all can be included in a set. I’ve seen this written as

S = {xP(x)}

There are a few ambiguities in this notation. It is not, I take it, defining a set as the one that includes all things, x,that have some property or other, P. Rather, it’s target is something like what I’ve already mentioned with Unrestricted Comprehension. It is saying that by any specifiable property P, a set can specify such that any and all things that have P are included in it as members of the set. That is, any property P can designate a set that has those and only those members. (Of course, sets are free to combine different properties in various ways, as well as limit the scope of such things as some of them in various ways, or all of them.)

One additional confusion that might arise is a problem inherent to the notion of a property. A property is typically not identified with the thing. For example, a red apple is an apple that has the property of red. The red of the apple is not the same as the apple as a whole. Similarly, being an apple might be thought to be a property of some thing. However, this is misleading if it is thought that being an apple is a property of something besides the apple itself. Here, the property must be identical with the thing itself, rather than the property of some additional thing. In short, a property P is to be understood as a catchall for both a predicative sense (x is red) and an identity sense (x is an apple).

Thus any specifiable property or thing at all may be included as a member of a set for so specifying it. You can have a set of all apples as well as a set of all red things. The things that can be included are also all sorts of things, from numbers, to physical objects, to sets themselves. Considering that that sets may specify any property or thing and thus included such things as members is an intuitive, simple idea. It was assumed to be true during the early days of set theory.

However, it was thought to have been obliterated by Russell’s Paradox. Because of Russell’s Paradox, sets became restricted in various ways, depending on the sort of set theory used, such that Russell’s Paradox could not be stated. The downside of this is that Unrestricted Comprehension was no more and that, to