I was painting with a broad brush and omitting talk of alternatives to Limited Comprehension, because my impression is that ZF (with or without C) is nowadays the standard background for doing set theory. Is there any other in use, not known to be straightforwardly equivalent to ZF? (ETA: I also elided all the history of how the subject developed leading up to ZF. For example, in Principia Mathematica Russell & Whitehead used some sort of hierarchy of types to avoid the inconsistency. That approach dropped by the wayside, although type systems are back in use in the various current projects to formalise all of mathematics.)
Restricted Comphrension prevents the question from even being asked. So, it “solves it” by removing the object from the domain of discourse.
Any answer to Russell’s question to Frege must exclude something. There may be other ways of avoiding the inconsistencies, such as the thesis that Adele Lopez linked, but one has to do something about them.
As far as I know for know, all of standard Mathematics is done within ZF + Some Degree of Choice. So it makes sense to restrict discussion to ZF (with C or without).
My comment was a minor nitpick, on the phrasing “in set theory, this is a solved problem”. For me, solved implies that an apparent paradox has been shown under additional scrutiny to not be a paradox. For example, the study of convergent series (in particular the geometric series) solves Zeno’s Paradox of Motion.
In Set Theory, Restricted Comprehension just restricts us from asking the question, “Consider this Set, with Y property” It’s quite a bit different than solving a paradox in my book. Although, it does remove the paradoxical object from our discourse. It’s really more that Axiomatic Set Theory avoids the paradox, rather than solve it.
I want to emphasize that this is a minor nitpick. It actually, ( I believe) serves to strengthen your overall point that RDT is an unsolved problem, I’m just adding that as far as I can tell—I think it’s safe to say this component of RDT ( self-reference) isn’t really adequately addressed in Standard Logic. If we allow self reference, we don’t always produce paradoxes, x = x is hardly, in any way self-evidently paradoxical. But, sometimes we do—such as in Russell’s famous case.
The fact that we don’t have a good rule system ( in standard logic, to my knowledge) for predicting when self-reference produces a paradox indicates it’s still something of an open problem. This may be radical, but I’m basically claiming that restricted Comprehension isn’t a particularly amazing solution for the self-reference problem, it’s something of a throwing the baby out with the bathwater kind of solution. Although, to its merit, that ZF hasn’t produced any contradictions in all these years of study- is an incredible feat.
Your point about, having to sacrifice to solve Russells question is well taken. I think it may be correct, the removal of something may be the best kind of solution possible. In that sense, restricted comprehension may have “solved” the problem, as it may be the only kind of solution we can hope for.
Adele Lopez’s answer was excellent, and I haven’t had a chance to digest the referenced thesis, but it does seem to follow your proposed principle- to answer Russells question we need to omit things.
I was painting with a broad brush and omitting talk of alternatives to Limited Comprehension, because my impression is that ZF (with or without C) is nowadays the standard background for doing set theory. Is there any other in use, not known to be straightforwardly equivalent to ZF? (ETA: I also elided all the history of how the subject developed leading up to ZF. For example, in Principia Mathematica Russell & Whitehead used some sort of hierarchy of types to avoid the inconsistency. That approach dropped by the wayside, although type systems are back in use in the various current projects to formalise all of mathematics.)
Any answer to Russell’s question to Frege must exclude something. There may be other ways of avoiding the inconsistencies, such as the thesis that Adele Lopez linked, but one has to do something about them.
As far as I know for know, all of standard Mathematics is done within ZF + Some Degree of Choice. So it makes sense to restrict discussion to ZF (with C or without).
My comment was a minor nitpick, on the phrasing “in set theory, this is a solved problem”. For me, solved implies that an apparent paradox has been shown under additional scrutiny to not be a paradox. For example, the study of convergent series (in particular the geometric series) solves Zeno’s Paradox of Motion.
In Set Theory, Restricted Comprehension just restricts us from asking the question, “Consider this Set, with Y property” It’s quite a bit different than solving a paradox in my book. Although, it does remove the paradoxical object from our discourse. It’s really more that Axiomatic Set Theory avoids the paradox, rather than solve it.
I want to emphasize that this is a minor nitpick. It actually, ( I believe) serves to strengthen your overall point that RDT is an unsolved problem, I’m just adding that as far as I can tell—I think it’s safe to say this component of RDT ( self-reference) isn’t really adequately addressed in Standard Logic. If we allow self reference, we don’t always produce paradoxes, x = x is hardly, in any way self-evidently paradoxical. But, sometimes we do—such as in Russell’s famous case.
The fact that we don’t have a good rule system ( in standard logic, to my knowledge) for predicting when self-reference produces a paradox indicates it’s still something of an open problem. This may be radical, but I’m basically claiming that restricted Comprehension isn’t a particularly amazing solution for the self-reference problem, it’s something of a throwing the baby out with the bathwater kind of solution. Although, to its merit, that ZF hasn’t produced any contradictions in all these years of study- is an incredible feat.
Your point about, having to sacrifice to solve Russells question is well taken. I think it may be correct, the removal of something may be the best kind of solution possible. In that sense, restricted comprehension may have “solved” the problem, as it may be the only kind of solution we can hope for.
Adele Lopez’s answer was excellent, and I haven’t had a chance to digest the referenced thesis, but it does seem to follow your proposed principle- to answer Russells question we need to omit things.