|
|
Re: Matheology § 154: Consistency Proof!
Posted:
Nov 25, 2012 3:19 AM
|
|
On 24 Nov., 21:34, Virgil <vir...@ligriv.com> wrote:
> In article
> <0ee396a8-91e7-4221-82e6-bed81af08...@o30g2000vbu.googlegroups.com>,
>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 23 Nov., 22:37, Virgil <vir...@ligriv.com> wrote:
>
> > > Analysis can show that the limit VALUE is oo in the extended reals, but
> > > does not presume to claim that there is a decimal, or any other place
> > > value based numeral, representing that limit value.
>
> > If the limit value
> > Limit[n-->oo] SUM[k=0 to n] a_k*10^k = oo
> > is accepted in the extended reals, then it is simply ridiculous to
> > claim that the abbreviation
> > ..., a_k, ..., a_3, a_2, a_1, a_0
> > is not in the abbreviations of the extended reals.
>
> The only such "abbreviations" in standard use for "extended reals are
> oo for the one point compactification or +oo and -oo for the two point
> compactification.
My proof does not need this abbreviation. My proof only needs the
facts
that the set { a_k | k in |N } of coefficients of the series is not
empty and the number |{ a_k | k in |N }| of coefficients of the
series is not zero. More is not required.
>
>
>
>
>
>
>
> > But William had agreed: "On the contrary, the fact that the analytic
> > *limit* cannot be described in terms of digits is the point."
>
> > And he stated proudly:
>
> > Analysis:
> > limit in real numbers: unbounded
> > (oo in extended reals)
> > limit of set of 1's: not estimated
>
> > Set Theory
> > limit in real numbers: not estimated
> > limit of set of 1's: {}
>
> > Therefore he would have to confess now that there is a contradiction
> > between set theory and analysis.
>
> To say that different definitions of "limit" can give different results
> only confuses WM, not anyone else.
The limit of a sequence is defined solely by the finite terms of the
sequence. Mathematics is the art of finding this limit, not the art of
inventing arbitrary limits.
Regards, WM
|
|
