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Having a universal set is a sufficient condition for Russel's paradox. That's not true. There are set-theories, e.g. Quine's NF [1] which allow universal sets,
by mafribe 11y ago
Having a universal set is a sufficient condition for Russel's paradox.
That's not true. There are set-theories, e.g. Quine's NF [1] which allow universal sets, and other things like the set of all ordinals, that are forbidden in ZF-style set-theories. The problem in ZF is caused by unlimited comprehension. NF circumvents this by restricting comprehension. Tom Forster [2] has written a great deal about set theories with universal sets, including the wonderful [3]. He makes the historical point that set theory was born with universal sets.
[1] http://plato.stanford.edu/entries/quine-nf/ http://plato.stanford.edu/entries/quine-nf/
[2] https://www.dpmms.cam.ac.uk/~tf/ https://www.dpmms.cam.ac.uk/~tf/
[3] T. E. Forster, Set Theory with a Universal Set. http://ukcatalogue.oup.com/product/9780198514770.do http://ukcatalogue.oup.com/product/9780198514770.do
- dvt 11y agoTrue, I was mostly referring to ZF-style set theories (which is what the thread is mainly about). Your point could even be extended by saying that there are proofs for a paraconsistent ZF with a universal set. Your [3] link doesn't work by the way, I'm interested in reading Forster!
- mafribe 11y ago[3] works on my browser. Anyway, the link was to the publisher's page for the book. Here is another one: http://www.amazon.co.uk/Set-Theory-Universal-Exploring-Universe/dp/0198514778 http://www.amazon.co.uk/Set-Theory-Universal-Exploring-Unive... .