Set Theory and Foundations of Mathematics

Re:Set Theory and Foundations of Mathematics by mafribe on Friday July 31, 2015 @02:22PM (#9983384)

   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/

[2] https://www.dpmms.cam.ac.uk/~tf/

[3] T. E. Forster, Set Theory with a Universal Set. http://ukcatalogue.oup.com/product/9780198514770.do

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Re:Set Theory and Foundations of Mathematics by dvt on Friday July 31, 2015 @02:34PM (#9983459)

True, 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!

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Re:Set Theory and Foundations of Mathematics by mafribe on Saturday August 01, 2015 @03:32AM (#9986271)

[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-Unive... .

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