Re: Proposal: 6NF

From: vc <boston103_at_hotmail.com>
Date: 12 Oct 2006 04:24:24 -0700
Message-ID: <1160652263.931558.180670_at_i42g2000cwa.googlegroups.com>


Jan Hidders wrote:
> vc wrote:
> >
> > Strictly speaking, Z is not a subset of R or even of Q (some stucture
> > is acquired some is lost when one moves between R, Q and Z), but the
> > subtlety is rarely important so one can say that Z is a subset of R (or
> > Q). A more correct way would be to say that Z is isomorphic to a
> > certain subset of R (or Z is ebedded in R).
>
> Hmm, wouldn't it be more correct to say that most mathematical
> definitions of these concepts either define them up to isomorphism (so
> the question doesn't really make sense)

There are many ways to construct reals in math, the rational "cuts" Q*, for example in the Dedekind implementation, are clearly not rational numbers as usually constructed from Z so one cannot say say that Q is a subset of R despite Q* being such subset. However, since Q* is isomorphic to Q, one can informally say so ;) . Similarly, one can treat reals as a "subset" of complex numbers such that the imaginary part is zero

>or define them explicitly as
> subsets (Z is the subset of R for which ...)?

That presupposes that you can construct reals from something other than Q (<- Z.). To be fair, there are R constructions, usually given as an exercise, where Z is an actual subset of R.

More interestingly, why bother with a trivial fact that Z may be [considered] a subset of R in whatever sense ? It's as unimportant as knowing that 1 is a subset of 2 in the von Neumann numerals although it is not in other constructions.

>
> > > I think
> > > the concept is called "specialization by constraint."
> >
> > What's that ?
>
> Defining a subclass A of B such that it is exactly the subset of B for
> which a certain predicate holds. It usually means that objects in A
> have no extra attributes except those already specified for objects in
> B.

Thanks.

>
> -- Jan Hidders
Received on Thu Oct 12 2006 - 13:24:24 CEST

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