Here’s Russell’s paradox (for those who are unfamiliar):

Let S be the set of all sets that don’t contain themselves.

If S contains itself, then S contains a set that contains itself, which defies the definition of S. On the other hand, if S doesn’t contain itself, then it is one of the sets that doesn’t contain itself, and therefore must contain itself.

That’s the paradox.

I wanted to know if Russell, or any one else, had ever solved it.

I just read about Russell’s paradox at SEP, and I don’t know if I’m completely satisfied by the type theory thing.

I seems to be saying that you can’t define a set by referring to itself, which is fair, but I can’t get around the fact that, at the end of the day, S is still a set that doesn’t contain itself, and therefore should be included.

Maybe the problem is with the word “all” in “All sets that don’t contain themselves”. Like SEP said:

So “all” in this case is not a universal “all” as in “All things that so happen to be sets that don’t contain themselves” but a particular “all” as in “All of these sets which don’t contain themselves” .

If this is the right interpretation of Russell’s Type Theory, then it seems the word “all” can never be universal - or that any universalization of “all” must not exceed the limits of a particular universe of elements that has been defined according to such limits.

I just ordered A Brief History of the Paradox from Amazon used. I browsed through it at a local bookstore and loved it. I love how you have to read paradox scenarios SO SLOWLY in order to really catch the loop or inconsistent parts. Truly fascinating! Bertie Russell was the man!

I think a problem with the very formation of Russell’s paradox is with the idea of an ALL set.
Once we are dealing with concepts like ALL or NOTHING; the infinite nature of these two tend to fuck things up from the get go.

Though infinite doesn’t necessarily mean “everything”, it is unlimited and causes serious problems.

After reading the Paradox book for a little while, I’m realizing that most paradoxes deal with some sort of conceptual infinite loop. Perhaps the words used to describe the paradox cause for an infinite loop, but the infinite nature of ALL and of ZERO or NOTHING tend to create these loops or paradoxes.

How it is the set of all reals thought to be limited?

Are we placing a limit on our meaning of ‘all’ - such that, say, extensions of the real numbers are excluded from the set of all reals, thus it is limited?

Then considered literally, it is determined to be infinite?

I follow, logically for sure - we’re saying one infinite set has ‘more’ than another in virtue of the latter being a subset of the former. This is along the same vein as other subsets of the real numbers.

But conceptually, because infinity is so elusive, it seems kinda dubious to declare one infinite set has more than another infinite set, regardless of their relationship.

Yeah, but everyone does, including mathematicians.

Infinite sets are purely theoretical. They serve theoretical purposes. They apply only to mathematics, in any useful sense.

My point is that we must keep on mind that sets occupy ranges. No set can be “everything” or it ceases to have meaning.

Another way to put this, is less mathematical terms, is that every term exists in a context (as does every function, which is closer to Russell’s exact point).

In terms of ‘things’ in the universe, however we conceive them (seconds, atoms, fractions of either, whatever) we can apply a number that is larger. But since numbers are an artificial system designed to enumerate things that do exist, we have to ask ourselves: does the very nature of infinity defy the thing to which we are applying it?

In the past, I’ve used differential equations as a defense of ‘free will’ inasmuch as such a flawed discussion can be had. Diff eqs have a finite number of solutions, but an infinite number of answers. So ‘free will’ can exist only insofar as it exists along the functions created by the solutions available. But what I am saying here is that these functions can’t really be said to go from negative infinity to infinity. But, of course, mathematics already tells us this since integrals can only be solved over a discreet distance.