I promised an explanation of Russell's paradox and how I believe it can be used outside of a purely mathematical context.
Interested parties should of course read the explanation on wiki or your favorite reference, but I'll give my informal (simplified) synopsis with what I see as the key features.
Naive set theory, as proposed in informal language, asserts Cantor's definition of set as "a gathering together into a whole of definite, distinct objects of our perception or of our thought." Although the logical language of set theory is first-order predicate logic, at this level of description we can use the simpler form of propositional logic: a set is well defined if we can formulate a propositional formula that describes its elements. That is, because the logical definition of a propositional formula P(x) is that for every x we can say (in principle) that P(x) is true or false but not both.
It follows from the assertion of this "definition" of set that the propositional formula P(x) iff x is a set is a well defined proposition - otherwise the notion of set would be poorly defined. But because P(x) iff x is a set is assumed to be a well formed proposition, by the definition of set it must define a set, i.e., the set of all things that are sets. At this point I'll note that we aren't really using many of the properties of sets. What we've really done is piggyback on the logical foundation to assert the existence of some collections of things that correspond with the action of propositions on them.
However, now that we have the set of all sets, we do need to assert properties of membership. Still we do not need the full machinery of Cantor's set theory to generate a paradox, only some rather intuitive rules about things that are members of collections. Specifically, if a collection of things or a set is well defined, and I take some of the members of that collection, the collection I have taken inherits the well-definition of the original collection. In talking about sets, a subset of a set is also a set.
And at this point we can also assert that the proposition of self-membership must be well defined, since the members of a set are definite. So whether a set has a particular member must be propositionally true or false, and since the set of all sets is itself a set, it follows that it is a member of itself.
Given rules of propositional logic, the negation of a proposition is also a proposition, so we can form a proposition S(x) iff x is not a member of itself, which derives from the negation of the proposition P(x) iff x is a member of itself - a proposition already established as well defined. Now we get another subset of the set of all sets, the set of all sets that are not members of themselves. This must be a set since it is a subset of the collection of all sets.
Note that the key move here is this assertion that the aggregate of all objects of a given type is itself an object of the same type.
At this point, the collection of all sets that are not members of themselves - In mathematical language, S = { X | X is not an element of X } - generates a paradox when we ask, is S a member of itself?
1. Start by assuming S is not a member of itself. In that case, it satisfies the definition of membership in S which is defined as the set of all sets that do not belong to themselves. That means S must be a member of S, i.e. it does belong to itself, which contradicts the assumption we started out with.
2. Suppose S is a member of itself. Then it must satisfy the conditions for membership in S which is defined as the set of all sets that do not belong to themselves. Thus we conclude that S must not be a member of itself, which again contradicts the assumption we started out with.
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Although much deeper predicate logic can be used to describe this, for the purposes of general discussion, this paradox does not really require much beyond:
1. Naive use of propositional logic in a free universe of consideration (notice we never scoped our universe of consideration to objects of a particular type - we were free to consider anything that might be encoded in a propositional formula).
2. A rudimentary notion of membership directly tied to propositional formulas.
Even the notion of self-membership (which is where things seem to start going off the rails) is necessitated by (1) and (2), which give us the assumption that the "collection of all collections" or set of all sets is well defined.
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So when I encounter mystical notions such as a God that is the "transcendent summation of all that exists", I see a very similar construction to naive set theory:
1. Is P(x) iff x exists a valid propositional formula? If not, we can stop right here and say our logical foundations are simply unusable. In this case, the statement that the God under discussion "exists" or perhaps does not is itself incoherent.
2. On the other hand, if we do (as most would) assume that existence is a well defined propositional property, then this God that contains the transcendent culmination of all that exists has properties very much like the set of all sets, and allows us to construct an analog of Russell's paradox.
To me, it's also a giveaway that type has become mixed here. If we postulate a thing that subsumes the entirety of existence, it seems suspiciously odd that this monstrous aggregation would have any properties in common with its lowly members. This is where the transcendent God appears to contradict a loving personal God. Suppose we define some other property such as P(x) iff x is capable of feeling love. Is the set or collection of all x that are capable of feeling love itself capable of feeling love? If we think it might, we're back in a set of all sets situation, and can very likely construct a paradox.
I suspect this prohibition on free mixing of type is a general principle. Can the free aggregation of all objects with a particular property itself be an object with that property? If so, it seems we'd be able to construct another Russell-type paradox.
What do you think? Is this paradox of use in general discourse? Or does it belong solely in Set Theory class?