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Re: set builder notation
Posted:
Aug 18, 2013 12:36 PM


On 8/18/2013 9:03 AM, dullrich@sprynet.com wrote: > On Sat, 17 Aug 2013 18:47:40 +0100, Peter Percival > <peterxpercival@hotmail.com> wrote: > >> dullrich@sprynet.com wrote: >>> On Sat, 17 Aug 2013 04:12:34 0700 (PDT), lite.on.beta@gmail.com >>> wrote: >>> >>>> >>>> S = {x /in A  P(x) } >>>> >>>> For the set builder notation above, what we really means is: >>>> >>>> all things x, such that "x is element of A *and* P(x) is true" correct? >>> >>> Yes and no. >>> >>> Yes: >>> >>> (1) {x in A  P(x)} >>> >>> is the same as >>> >>> (2) {x  x in A and P(x)}. >>> >>> No: >>> >>> No, because (2) is actually not a "legal" >>> construction of a set! (2) is of the form >>> >>> (3) {x  Q(x)}, >>> >>> and things of the form (3) are officially not >>> allowed. >>> >>> Not allowed because they lead to contradictions: >>> Let >>> >>> S = {x  x is not an element of x}. >> >> But is 'x is not an element of x' of the form 'x in A and P(x)'? > > No. > >> I >> suppose what I'm doing is challenging you to reproduce Russell's paradox >> with sets of the form {x  x in A and P(x)}. I'm rather sure (but I >> know nothing) that {x in A  P(x)} is *nothing but* alternative notation >> for {x  x in A and P(x)}. > > Informally it's just an alternate notation. > > The fact that {x  P(x)} is not allowed seems more important. > If you want you can change things so that {x  P(x) is > allowed as long as P(x) is of the form "x in A and Q(x)". >
Yes.
One can find a discussion of a language in this form in "Topological Model Theory" by Flum and Zeigler
http://books.google.com/books/about/Topological_Model_Theory.html?id=YpkfAQAAIAAJ
The introduction of restricted quantifiers via the formation rules are also constrained relative to equivalences with negation normal forms.



