Re: naive question from a non-mathematician


Stephen Montgomery-Smith wrote:
Gene Ward Smith wrote:
cody.roux@xxxxxxxxx wrote:


Yes. It is usually implicitly assumed when you are talking about the
field of complex numbers, that you have the inclusions N in Z in Q in R
in C. It probably would be more correct (and horribly unpractical) to
talk about injective mappings in each case.


I don't see why; all the above are instrinsically defined in terms of C.

Like I said in a different post, it depends upon your framework. Most
modern mathematicians define everything in terms of sets and set theory.

If I were a mathematician, I would most defnitely choose to be
old-fashioned!

Thus the natural numbers are, more or less, defined as the set of
finite ordinals, the integers are pairs of natural numbers quotiented
out by the equivalence relation (a,b)~(c,d) iff a+d=b+c, the rationals
are pairs of integers (the second being non-zero) quotiented out by
another approprate equivalence relation, the reals are constructed from
the rationals usually either by Dedekind sections, or by some quotient
of the cauchy sequences, and the complex numbers are pairs of real numbers.

But then again, maybe the complex numbers aren't really an uncountable
set, but actually a tuple (C,+,*), where C is the underlying set and +
and * are the binary operations of addition and multiplication. Or
maybe you want an additional element in your tuple - the collection of
open sets.

There are just so many ways of defining these objects in set theory, and
so to say R is a subset of C just doesn't cut it if you are going to be
nit-picking.

Now one could argue that this shows that our modern rigorous definitions
of math aren't an adaquate description of how we really do math, but
then I wouldn't disagree.

Stephen

.

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