Re: On Formal IS-A definition
Date: Sat, 8 May 2010 19:46:50 -0700 (PDT)
Message-ID: <226cbf22-146b-4e4f-8013-a35f27284332_at_v29g2000prb.googlegroups.com>
On May 8, 5:51 pm, Erwin <e.sm..._at_myonline.be> wrote:
> It is true that mathematical set theory has no constraint on what type
> of thing can or cannot be member of a set.
Modern axomatic set theory has no concept of types. However it is normally a requirement that every element of a set is a set.
Set theorists distinguish between classes and sets. A class is a collection of sets and proper classes cannot be members of sets. This is an example of where Naive Set Theory is too simplistic.
The idea to treat elements of sets in an opaque way is just a convenience. E.g. Peano axioms and Set theory are compatible. It just means that each integer value is some unspecified set. That is in keeping with the spirit of the axiomatic approach. The integers are defined uniquely up to isomorphism by the axioms. Received on Sun May 09 2010 - 04:46:50 CEST