🚀 go-pugleaf

On 27 Mrz., 05:45, Barb Knox <s...@sig.below> wrote:
> In article <iml34b$k0...@charm.magnus.acs.ohio-state.edu>,
>  Jim Burns <burns...@osu.edu> wrote:
>
>
>
> > On 3/24/2011 8:05 PM, Albrecht wrote:
> > > On 24 Mrz., 15:08, James Burns<burns...@osu.edu>  wrote:
> > >> Albrecht wrote:
> > >>> On 24 Mrz., 10:31, William Hughes
> > >>> <wpihug...@gmail.com>  wrote:
> > >>>> Absolute piffle.  There is no reason that defining the
> > >>>> existence of the limit of finite ratios means that you
> > >>>> have to define either infinite quantities, or the ratio
> > >>>> of infinite quantities.
>
> > >>> So you admit that an infinite decimal number has crucial
> > >>> other properties than a finite decimal number? You admit
> > >>> that a number like 3.1415... is crucial different from a
> > >>> number like 0.75? But you insist that the "behaviour" of
> > >>> a number like 3.1415... should be like the "behaviour" of
> > >>> a number like 0.75?
>
> > >>> The axioms behind this view are silly.
>
> > >> If I am not mistaken, there is only one axiom that
> > >> distinguishes infinite decimal numbers (real numbers)
> > >> from finite decimal numbers (rational numbers, or close
> > >> enough for our purposes).
>
> > >> <  For every non-empty set of real numbers, if it has
> > >> <  an upper bound, then it has a least upper bound.
>
> > >> Upper bound and least upper bound are defined in the
> > >> expected ways:
>
> > >> Upper bound of set A == a real number greater than or
> > >> equal to every element of A
>
> > >> Least upper bound of set A == an upper bound of A
> > >> less than or equal to every upper bound of A
>
> > >> I just wanted to be sure:
> > >> do you find this axiom silly?
>
> > > No. I have the axiom of infinity in mind. And the
> > > combination of the power set axiom with the axiom of
> > > infinity. Without these axioms the interpretation of
> > > the diagonal argument of Cantor is completely
> > > different from the common interpretation.
>
> > Yes, very different. No set of real numbers, for example.
> > (Well, no proof that the real numbers exist, unless
> > you add another axiom saying all sets are finite.)
> > Also, no set of rational numbers, no set of integers,
> > no set of naturals. Very different.
>
> > Are you saying, then, that pretty much all of
> > everyday mathematics is silly? Or do you see parts
> > that do not look silly?
>
> > ---
>
> > Please consider an argument for the not-silliness
> > of the completeness of the reals (see Least Upper Bound
> > Axiom) which I suspect is a better suspect for the
> > origin of your chuckles.
>
> > When any calculation much more difficult than
> > "2 + 2 = 4" is done, very often it will proceed in
> > stages, "successive approximations" it is often called.
> > Even something as simple as dividing 7 into 3
> > (longhand) produces a digit, then another, then another
> > in an unending stream.
>
> > This series of partial results, these successive
> > approximations, form a Cauchy series: all the
> > partial results agree with each other to any
> > desired degree after a certain point in the
> > series. None of this so far needs real numbers
> > in order to be true.
>
> > A Cauchy series is a Cauchy series whether or not
> > there is anything that the series converges to.
> > After all, all we are given is that the partial
> > results get close to /each other/. But, it would be
> > convenient if there were a single /thing,/ call it
> > what you will, that all of the partial results
> > draw close to, the limit of the series.
>
> > And, then, there is the question of alternative
> > methods of calculation. What does it mean to have
> > two ways of calculating "the same number" if none
> > of the partial results agree at any stage? The
> > "same number" seems to mean that the partial results
> > from one series agrees with the partial results
> > from the other series to any desired degree if
> > one follows the two series out far enough.
>
> > One way of building the set of real numbers is
> > to collect all the Cauchy series of rational
> > numbers that "calculate the same number" and
> > identify the collection as a real number and
> > the limit of those Cauchy series. (Rational
> > numbers already exist and pose no logical
> > problems.)
>
> >  From this identification of a real number as
> > a collection of agreeable Cauchy series follows
> > two very desirable properties:
> > (1) Every series of successive approximations
> > converges to an answer.
> > (2) If two series of successive approximations
> > get arbitrarily close, then they converge
> > to the same answer.
> > If these properties seem like common sense, it is
> > because we have all grown up using real numbers.
> > They are not inevitable.
>
> > If you have an order field ('+', '-', '*', '/',
> > and '<' with their usual meanings) that satisfies
> > properties 1 and 2, then you have the real numbers,
> > with all of its silly consequences.
>
> > It may be possible to devise an alternate system,
> > the not-silly-real numbers, if you like. In order
> > to do that, though, you will have to give up
> > something. Are you sure that you want to?
>
> Whatever he does next, he will be sure that it is correct.
>
> Your exposition is helpful and to the point, but it will not get
> through.  
>
> --
> ---------------------------
> |  BBB                b    \     Barbara at LivingHistory stop co stop uk
> |  B  B   aa     rrr  b     |
> |  BBB   a  a   r     bbb   |    Quidquid latine dictum sit,
> |  B  B  a  a   r     b  b  |    altum videtur.
> |  BBB    aa a  r     bbb   |  
> -----------------------------

You must be a sybil. You love this job?

Albrecht