🚀 go-pugleaf

On 26 Mrz., 18:09, 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?
>


Pritty much all of everyday mathmatics is well done with proper
classes of naturals, rationals, reals, ...

Best
Albrecht