By the way, quick history of Russell’s paradox and related matters, with possible application to the original topic. :)
Russell first pointed out his namesake paradox in Frege’s attempt to axiomatize set theory. So yes, it was a mathematical statement, and, really, it’s pretty simple to state it. Why nobody noticed before this paradox before then, I have no idea, but it does seem to be worth noting that nobody noticed it until someone actually attempted to sit down and actually formalize set theory.
However, Russell was not the first to notice a paradox in naïve set theory. (Not sure to what extent you can talk about paradoxes in something that hasn’t been formalized, but it’s pretty clear what’s meant, I think.) Cesare Burali-Forti noticed earlier that considering the set of all ordinals leads to a paradox. And yet, despite this, people still continued using naïve set theory until Russell! Part of this may have been that, IIRC, Burali-Forti was convinced that what he found could not actually be a paradox, even though, well, in math, a paradox is always a paradox unless you can knock out one of the suppositions. I have to wonder if perhaps his reaction on finding this may have been along the lines of ”...but somebody would have noticed”. :)
And note also that even Russell’s paradox was not phrased originally in this way. His original phrasing as I understand it rested on taking the set of all sets A and then looking at the cardinality of that set’s powerset P(A). Then we have |P(A)| > |A| but P(A) ⇐ A so |P(A)| ⇐ |A| which is a contradiction. This is essentially the same as Russell’s paradox when one expands out the details (particularly, the details in the proof that in general a set has cardinality strictly less than its powerset).
By the way, quick history of Russell’s paradox and related matters, with possible application to the original topic. :)
Russell first pointed out his namesake paradox in Frege’s attempt to axiomatize set theory. So yes, it was a mathematical statement, and, really, it’s pretty simple to state it. Why nobody noticed before this paradox before then, I have no idea, but it does seem to be worth noting that nobody noticed it until someone actually attempted to sit down and actually formalize set theory.
However, Russell was not the first to notice a paradox in naïve set theory. (Not sure to what extent you can talk about paradoxes in something that hasn’t been formalized, but it’s pretty clear what’s meant, I think.) Cesare Burali-Forti noticed earlier that considering the set of all ordinals leads to a paradox. And yet, despite this, people still continued using naïve set theory until Russell! Part of this may have been that, IIRC, Burali-Forti was convinced that what he found could not actually be a paradox, even though, well, in math, a paradox is always a paradox unless you can knock out one of the suppositions. I have to wonder if perhaps his reaction on finding this may have been along the lines of ”...but somebody would have noticed”. :)
And note also that even Russell’s paradox was not phrased originally in this way. His original phrasing as I understand it rested on taking the set of all sets A and then looking at the cardinality of that set’s powerset P(A). Then we have |P(A)| > |A| but P(A) ⇐ A so |P(A)| ⇐ |A| which is a contradiction. This is essentially the same as Russell’s paradox when one expands out the details (particularly, the details in the proof that in general a set has cardinality strictly less than its powerset).
Ah, good point. I’d forgotten about that part. IIRC he first noted that and then expanded out the details to see where the problem was.