Re: JSH: Keep it simple

William Hughes wrote:
> jstevh@xxxxxxx wrote:
> > William Hughes wrote:
> >
> > <deleted>
> >
> > > It doesn't matter if you have a rational or irrational. It is easy to
> > > show that if a ring R
> > >
> > > a: contains the integers
> > > b: contains an element between 0 and 1
> > > c: contains all infinite sums or elements of the ring
> > >
> > > then R must contain all real numbers. In particular if the algebraic
> > > integers contain all infinite sums of algebraic integers then the
> > > algebraic integers contain all reals.
> > >
> > > -William Hughes
> > >
> >
> > Specious claims which rely on "moving the dot" in the decimal system.
> >
> > For instance, you may claim that you can get infinitely close to 1/2
> > with all those elements, except that you might write something like
> >
> >
> > 0.414
> >
> > which is 414/1000 without using the decimals, so anything you write
> > that has decimal places is just a fraction.
> >
> > But if you don't have fractions, then you don't have decimal places.
> >
> > Understand?
> >
> > Let me repeat, the decimal system is just a way to express a fraction.
> >
> > So if you don't have the decimal system, how do you propose to get
> > "infinitely close" to any particular fraction?
> >
> > How do you show it?
> >
> > You can't. I suggest you try, and remember, no decimal places--no
> > fractions to try and get close to fractions.
>
> Consider
>
> (n^2 + n)^(1/2) -n
>
> this can be made arbitrarily close to 1/2 by taking n large enough.
>
> -William Hughes
> >

Yup, but arbitrarily close does not give you 1/2.

Rings like the ring of algebraic integers do not have 1/2 not because
of the reasons you think: exclusion of infinite convergent sums, as
they do NOT exclude them.

So there must be another reason, right?

The exclusion is done for the "field of rationals" and for appending a
fraction like 1/2 to the ring of integers, so it's an arbitrary thing.

If you don't exclude infinite convergent sums then you don't have just
rationals--you get the entire field of reals.

So people arbitrarily do this exclusion thing for specific cases, which
is why the theory of ideals fails, as what follows logically doesn't
give a damn about human preference.

The mathematics is not human.

So people come up with bogus ideas, like sqrt(4) is just 2 and not 2 or
-2 and the mathematics doesn't shift, so if you try to say only one
exists you can end up with some mathematical argument that predicts a
wrong result, so the numbers won't support you.

But people can do that quirky thing of choosing to think one way when
it suits them and then thinking another when they want, so they can be
what correct mathematics is not--inconsistent.


James Harris

.

Relevant Pages

  • Re: JSH: Keep it simple
    ... > integers contain all infinite sums of algebraic integers then the ... > algebraic integers contain all reals. ... which is 414/1000 without using the decimals, ... the decimal system is just a way to express a fraction. ...
    (sci.math)
  • Re: JSH: Keep it simple
    ... >> integers contain all infinite sums of algebraic integers then the ... >> algebraic integers contain all reals. ... > which is 414/1000 without using the decimals, ... the decimal system is just a way to express a fraction. ...
    (sci.math)
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