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Topic: I rarely make silly mistakes, but Euler made a huge blunder in S
= Lim S

Replies: 6   Last Post: Oct 6, 2017 3:42 PM

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genmailus@gmail.com

Posts: 323
Registered: 8/18/16
Re: I rarely make silly mistakes, but Euler made a huge blunder in S
= Lim S

Posted: Oct 4, 2017 3:13 PM
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On Wednesday, 4 October 2017 15:09:33 UTC-4, Markus Klyver wrote:
> Den onsdag 4 oktober 2017 kl. 19:29:16 UTC+2 skrev John Gabriel:
> > On Wednesday, 4 October 2017 11:37:26 UTC-4, Me wrote:
> > > On Wednesday, October 4, 2017 at 12:43:52 PM UTC+2, genm...@gmail.com wrote:
> > >

> > > > Even "Me" has finally understood that my definition is a D. Cut.
> > >
> > > No, I haven't. Sorry about that.
> > >
> > > But... you write:
> > >

> > > > L={-1 < x < pi} and R={pi < x < 4} where x \in Q
> > >
> > > Again, a rather "uncommon" notation (to say the least).
> > >
> > > For example there seems to be a free variable, "x", in the expression "{-1 < x < pi}" (for example). Hence I don't think it qualifies for a "term" just denoting a "specific" set. Moreover you "externalize" the information that "x" ranges over all elements in Q; we usually put this into the "set terms" (such that they are "self-supporting" (selfcontained)).
> > >
> > > Hence I guess that you actually meant to write:
> > >
> > > L = {x e Q : -1 < x < pi}
> > > and
> > > R = {x e Q : pi < x < 4} .
> > >
> > > Actually this corresponds to a quite natural way of referring to these sets.
> > >
> > > For example, {x e Q : -1 < x < pi} is /the set of all elements in Q that are larger than -1 and smaller than pi/.
> > >
> > > Now concerning Dedekind cuts, you might improve your approach by just defining:
> > >
> > > L = {x e Q : x < pi}
> > > and
> > > R = {x e Q : pi < x} .
> > >
> > > Then (L, R) would actually qualify for a "D. cut", I guess.
> > >
> > > So why not choose the simpler approach?

> >
> > Any cut of the form
> >
> > (m, k) U (k, n) where m < k and k < n
> >
> > is EQUIVALENT to
> >
> > (-oo, k) U (k, oo) where k is not a rational number.
> >
> > The tail parts (-oo,m) and (n, oo) which are discarded, are irrelevant. In fact, the tail parts do not feature in my disproof of the D. Cut and can be added in at any time without any loss of generality.

>
> No, those sets are not the same set.


They are the same set:

(m, k) U (k, n) is a subset of

(-oo,m) U (m, k) U (k, n) U (n, oo)

I only work with the subset in my proof and I can choose any subset I like without the tail parts affected at all.



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