I've read 'Axiomatic Set Theory' by Patrick Suppes, and one thing I've noticed throughout is that he seems to be obsessed with definitions, and he tries to allow for urelements. Is this standard for ZFC?

I thought in general when we say 'set' in ZFC we really mean 'pure set', and so avoid having to worry about individuals. In addition I've never seen such a fuss over definitions in any other mathematical book I've read, is this something I should get used to in Set Theory?

If this is not standard, can anyone direct me to a book similar to Suppes' which builds (from the axioms) all the usual set theoretical structures needed for other areas of mathematics that is?

  • $\begingroup$ Only a comment - if you forbid ur-elements, you are nor more allowed to use examples like : the set $\{ John, Jim \}$ ... unless you show how to "build up" e,g, $John$ starting from $\emptyset$. $\endgroup$ – Mauro ALLEGRANZA Sep 12 '15 at 15:48

From Wikipedia:

The Zermelo set theory of 1908 included urelements. It was soon realized that in the context of this and closely related axiomatic set theories, the urelements were not needed because they can easily be modeled in a set theory without urelements. Thus standard expositions of the canonical axiomatic set theories ZF and ZFC do not mention urelements (For an exception, see Suppes).

  • $\begingroup$ Well that answers the first part, it isn't standard, can you direct me to a similar book which is? $\endgroup$ – Nethesis Sep 12 '15 at 15:19
  • $\begingroup$ I think a good introduction (though it may leave you unsatisfied if you would like to see more rigour in the statement of axioms) is Paul Halmos' Naive Set Theory. It does not treat ur-elements. $\endgroup$ – parsiad Sep 12 '15 at 15:20
  • $\begingroup$ Mmm, I'm more looking for an axiomatic approach, thanks though $\endgroup$ – Nethesis Sep 12 '15 at 15:21
  • $\begingroup$ Paul Halmos' book is axiomatic; the axioms are stated in plain language (hence the prefix "Naive"). $\endgroup$ – parsiad Sep 12 '15 at 15:22
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    $\begingroup$ Jech, Hrbeck - Introduction to Set Theory is good as well as Enderton - Elements of Set Theory. $\endgroup$ – Tim Raczkowski Sep 12 '15 at 15:23

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