Re: naive question from a non-mathematician

On Sun, 28 May 2006 15:51:47 GMT, Stephen Montgomery-Smith
<stephen@xxxxxxxxxxxxxxxxx> wrote:

Stephen Montgomery-Smith wrote:
Gene Ward Smith wrote:

[...]

However, I do think that defining things axiomatically is often a good
way to do things. For example, to define the real numbers as "a
complete ordered field" is precisely right, because it captures the
essence of what we mean by the real numbers.

That's certainly the right way to think of the reals. I object
to calling it a "definition" - before we can call it a definition
we need to either assume that a complete ordered field exists
(as is often done in "advanced calculus" courses, explicitly
or not) or construct one from something else we're assuming,
like maybe set theory.

But I don't think that the way you define the complex numbers should be
considered the proper way. It is the "cardinality the continuum" part
that seems very unnatural to me. Then I thought - the complex numbers
is really the algebraic closure of the reals. But even that isn't how
we actually define or think of them - algebraic closure of the compelx
numbers is usually presented as a theorem, not a definition. No, the
proper definition is "the reals extended by adding in a square root of -1."

But I do think that your way of defining Q and Z is right.

On the other hand, the proper way to define N is not as the positive
elements of Z, but as an object satisfying Peano's axioms.

Of course, understand that phrases like "precisely right" and "the
proper way" are not meant as absolute statements, but rather as
statements of taste.

Stephen


************************

David C. Ullrich
.

Relevant Pages

  • Re: naive question from a non-mathematician
    ... modern mathematicians define everything in terms of sets and set theory. ... But I don't think that the way you define the complex numbers should be considered the proper way. ... It is the "cardinality the continuum" part that seems very unnatural to me. ... Then I thought - the complex numbers is really the algebraic closure of the reals. ...
    (sci.math)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... >> for the silly reason that M doesn't know about all the reals. ... >> naturals of M and those of N, and if M has any nonstandard naturals ... means M has a wellordering of the reals whose every proper initial segment is ... That wellordering can be coded by a set of reals. ...
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  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... >> for the silly reason that M doesn't know about all the reals. ... >> naturals of M and those of N, and if M has any nonstandard naturals ... means M has a wellordering of the reals whose every proper initial segment is ... That wellordering can be coded by a set of reals. ...
    (sci.logic)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... >> for the silly reason that M doesn't know about all the reals. ... >> naturals of M and those of N, and if M has any nonstandard naturals ... means M has a wellordering of the reals whose every proper initial segment is ... That wellordering can be coded by a set of reals. ...
    (comp.theory)
  • Re: general discussion
    ... minus one is making *two* implied ontological ... closure of the reals. ... same useful sense that C is 'the' algebraic closure of R: ... or 'complex plane': ...
    (rec.arts.sf.composition)