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Having a universal set [in naive set theory] is a sufficient condition for Russel's paradox. See Naive Set Theory[1] bottom of page 6. This is why we can have n

by dvt 11y ago

Having a universal set [in naive set theory] is a sufficient condition for Russel's paradox. See Naive Set Theory[1] bottom of page 6. This is why we can have no Universe in any consistent set theory. Edit: @mafribe makes the point that there are some set theories that can still have universal sets by culling other features that ZF-style set theories have. I was mostly referring to ZF-style set theory (hence my citation). Indeed, one could even make a ZF-style set theory paraconsistent and still have Universal sets. [1] http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Alexandria/Logica%20y%20Conjuntos/Paul%20R.Halmos%20-%20Naive%20Set%20Theory.pdf http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Al...

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 ago
  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!
  - 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... .
Shaniqua 11y ago
You can fix the Russel's Paradox in ZF as well.

by dvt 11y ago

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 ago
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!
- 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... .

Shaniqua 11y ago

You can fix the Russel's Paradox in ZF as well.