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I thought this would be about Skolem's Paradox, which _really_ broke set theory, and which set theorists have done their very best to ignore and handwave away s
by A_D_E_P_T 3y ago
I thought this would be about Skolem's Paradox, which _really_ broke set theory, and which set theorists have done their very best to ignore and handwave away since its introduction.
- AnimalMuppet 3y agoThe Wikipedia article (https://en.wikipedia.org/wiki/Skolem's_paradox) https://en.wikipedia.org/wiki/Skolem's_paradox) says that 1) it is "not an actual antinomy like Russell's Paradox" (that is, it's an apparent contradiction rather than a real one), and 2) "the result quickly came to be accepted by the mathematical community" (even though Zermelo was highly opposed to it). So I think your claim is rather misleading.
- A_D_E_P_T 3y agoOh it's much worse for set theory than an actual antimony. For most of those are mere word-games of the "does not compute" variety, whereas Skolem's Paradox states plainly, and uncontrovertibly, that there's no such thing as an absolutely uncountable infinite set. All sets have countable models, however uncountably vast they may be. It thus completely explodes all of Cantor's work on "countable" and "uncountable" infinities, rendering all infinite sets countable and equal. (As should have been intuitively obvious to begin with.) And it's ignored because people still find the nesting hierarchies of set theory useful, on occasion, even though what they now mean is very much open to question. Most of the people who have famously disparaged set theory as a theoretical foundation for mathematics -- like Wittgenstein -- have done so because, after Skolem, the whole thing is quite uncertain.