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According to my limited understanding, the axiom schema of unrestricted comprehension states that, for any predicate, there exists a set whose members are exactly those that satisfy the predicate. Russell showed this to be false by considering the set of all sets that do not contain themselves, and finding that if that set did not contain itself it implied that it did, and vice versa, leading to a contradiction. This has been resolved in several ways, but the most common is to dump the idea more or less completely, and as a result, prevent the existence of a universal set. Why don’t we instead just modify the axiom?

We could instead say that, for any predicate, there exists a set that contains all objects that satisfy that predicate. In such a set, we allow for other objects to be contained in the set. Let’s try to use Russell’s Paradox on it. Take a set containing all sets that do not contain themselves. We allow this set to contain other objects. If the set does not contain itself, then it should, which means it must contain itself. If it does contain itself, then it no longer satisfies the predicate of not containing itself, but it can still be in the set under our definition, so we’ve eliminated the paradox. This also seems to resolve the Burali-Forti Paradox and Cantor's Paradox, so it seems like there’s no longer anything in the way of the universal set.

Now, there is a strange notion to this idea, which is that, simply by putting all sets that do not contain themselves into a set, we get a set that does contain itself chucked in for free automatically. At first, I was uncomfortable with this idea, but I no longer feel that way. We consider the set containing elements a, b, and the set containing a and b, to be different from the set containing elements a and b. That is, we consider {a, b, {a, b}} to be different from {a, b}. But why do we do this? Clearly, {a, b} contains both a and b, and intuitively contains the collection that is a and b put together. It seems we treat a set as more than this collection, we treat a set as this collection “in a box”, that is, we treat the set as the collection taken in isolation, so that {a, b} does not contain itself. Why don’t we keep this notion of a collection in a box so that things like a topology still make sense, but say that a set automatically contains all of its subsets, outside of boxes? This seems to make sense, and if we allow it, then all sets contain themselves. The set of all sets that do not contain themselves is then the null set, but more importantly, contains itself. We could make an axiom that, for any predicate, there exists a set that contains exactly all objects that satisfy that predicate and all unions of those objects.

For what reason is this wrong?

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  • $\begingroup$ "The set of all sets that do not contain themselves is then the null set, but more importantly, contains itself." - This is an obvious contradiction, unless you stumbled over your phrasing. Also, this is several questions at once, which is against best practices for this site. $\endgroup$ – Malice Vidrine Apr 4 '17 at 13:07
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    $\begingroup$ "Why don’t we use a weaker version of the Axiom Schema of Unrestricted Comprehension?" We use it : Axiom of Separation (or Specification) stating that "for any set and for any predicate, there exists the subset of that set whose members are exactly those that satisfy the predicate." $\endgroup$ – Mauro ALLEGRANZA Apr 4 '17 at 13:07
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In other words, you're proposing to weaken the universal comprehension axiom to

For every predicate $p$ there exists a set $X$ such that $X$ comprises at least every $y$ where $p(y)$ is true (but may have more elements than that).

You're right that this wouldn't be paradoxical. But that is because it is extremely weak. This axiom is satisfied for every $p$ the moment there exists a set of all things. (And a set of all things would have to exist the moment we consider your axiom with $p(x)$ taken as $x=x$).

This means that the axiom is practically useless. When we use sets while doing ordinary mathematics, we don't merely want our sets to have enough elements -- it is usually critically important that the sets contain only the elements we want them to -- and your proposal doesn't leave us any way to construct such a set.

Usually we'd use Zermelo's axiom of separation to remove unwanted elements from a set, but if we take that together with your proposal, then Russell's paradox still results immediately.


Remember that it's not in itself an advantage of a proposal that it would allow us to write down some combinations of symbols that we previously didn't have a meaning for (e.g., "$\frac 00$" is a popular candidate for this). It only represents progress if the new thing we can write down represents a meaning that we have a reason to want to express.

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If there is not a contradiction in this system, it's because, as Henning Makholm states, it's extremely weak. To see how useless this would be, consider that we get induction over the natural numbers because we can form a set whose members are exactly those that are in every set containing $0$ and closed under successor; without this, induction simply doesn't get off the ground. And it's not just natural numbers; any kind of induction is going to be stunted.

Now, separation is a restriction of set abstraction but it seems like you're interested in something else. If you want to keep a universal set, there are several options that are well studied and not outright insane. The big ones are NFU and GPK. The former restricts abstraction to those formulae which are "stratified"; formulae which could be well-formed formulae of simplified Russellian type theory save that you've carefully chucked the typing discipline. The latter restricts abstraction to positive formulae (those in which negation and implication don't appear), with some other gear that has a topological motivation.

Both of these alternatives are strange places, but at least we can still do things like analysis, algebra, and number theory.

I cannot address the second question, as I'm not sure it makes sense. I will note that it's consistent with NFU to have sets such that $a=\{a,b\}$ (and hence $\{a,b,\{a,b\}\}=\{a,b\}$); even sets such that $a=\{x:a\in x\}$. But the truth is they're not very exciting objects, except as counterexamples to see how badly Foundation can fail.

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