Re: JSH: Critique means slow, and thorough

W. Dale Hall wrote:
> jstevh@xxxxxxx wrote:
> > William Hughes wrote:
> >
> >>jst...@xxxxxxx wrote:
> >>
> >>>Adjoining such an element gives you the field of reals.
> >>>
> >>>Now this issue has come up before, where I've noted that even
doing
> >>>something as simple as adjoining 1/2 to the ring of integers will
> >>
> >>give
> >>
> >>>you reals.
> >>>
> >>
> >>And it was immediately noted that this is nonsense as
> >>rings are not closed under limits (you don't even
> >>need a definition of limit). Rings do not have
> >>infinite sums.
> >>
> >> - "William Hughes"
> >
> >
> > That is an unprovable assertion, as you cannot block the infinite
sums.
> >
>
> It's not any such thing. The definition of the process of adjoining
> an element to a ring is that the result is the *minimal* ring that
> contains both the original ring and the new object. Since the
definition
> of *ring* is satisfied by only using finite sums, the restriction to
> finite sums is justified by *minimality*.

It doesn't work.

That is, claiming it's the minimal ring simply doesn't have
mathematical reality with an infinite sized ring.

> Have you noticed that the field of rational numbers contains Z[1/2]?
>

Yup. And I've previously posted on the subject.

> Why haven't you been railing about the "flaw" in the rational
numbers?
> Pi is rational!
>

No, pi is not rational.

Again, for those who wonder what the latest discussion is about, if you
append 1/2 to the ring of integers, you get the reals--if you allow
infinite sums.

Well, for some time, the standard opinion has been that you can simply
declare that you do not allow infinite sums, like by saying you're
using a minimal ring.

I say that doesn't work and I can prove it, as my own research proves
it.

You may think you have a minimal ring, but as your qualification has no
mathematical meaning, you get the full ring, which is a field, and it's
the field of reals.

Now that's a big deal as people get taught that you can get all these
rings in-between integers and reals by appending various fractions,
like you have

Z[1/2]

and I'm saying, you can't, if the rings are infinite sized, as in fact,
what you get is reals, and the mathematics behaves that way.


James Harris

.

Relevant Pages

  • Re: JSH: Critique means slow, and thorough
    ... >>>Adjoining such an element gives you the field of reals. ... > to limits, you are correct in pointing out that infinite summation ... You have a set of ring axioms that talks about infinite sums? ... the ring of real numbers does not have infinite sums. ...
    (sci.math)
  • Re: JSH: Critique means slow, and thorough
    ... Aring is closed under addition. ... What does it mean to "block the infinite sums"? ... Dale Hall wrote:) ... The definition of the process of adjoining ...
    (sci.math)
  • Re: JSH: Keep it simple
    ... R is closed under convergent infinite sums, ... First note that since R is a ring, ... So we only need to show that R is dense in the reals. ...
    (sci.math)
  • Re: JSH: Critique means slow, and thorough
    ... and the algebraic integers is the ordinary integers Z. ... infinite sums, so what happens is by *saying* they are blocked, you step into a non-math area, which falls apart if you do anything that depends on your arbitrary choice. ... you need to take the step *explicitly* to define what it is you mean by a convergent series. ... you need to prove that any ring must be complete with respect to that definition of convergence. ...
    (sci.math)
  • Re: JSH: Critique means slow, and thorough
    ... integers gives you reals, as you end up with numbers that result from infinite sums, whether you want them or not. ... The mathematical definition of a commutative ring is well known, ... where 1 and -1 are the only rational units is the field of reals, and if i is in the ring, then you have the field of complex numbers. ... and more likely to be a ruse you pull out to derail any further discussion regarding my proof that one can take an arbitrary algebraic number, and establish it as a ratio a/b of algebraic integers for which one can find algebraic integers u and v that prove a and b to be ...
    (sci.math)