Date: Oct 4, 2017 2:16 PM
Author: bursejan@gmail.com
Subject: Re: I rarely make silly mistakes, but Euler made a huge blunder in S<br> = Lim S

I guess you don't understand what it means to

construct something, either in the Dedekind

sense or in the Cauchy sense.

What might help your little bird brain is

maybe the notion of bootstrapping something.

https://en.wikipedia.org/wiki/Bootstrapping#Computing

All real construction are Münchhausen feats:

https://en.wikipedia.org/wiki/Bootstrapping#/media/File:Zentralbibliothek_Solothurn_-_M%C3%BCnchhausen_zieht_sich_am_Zopf_aus_dem_Sumpf_-_a0400.tif

Am Mittwoch, 4. Oktober 2017 19:29:16 UTC+2 schrieb 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.