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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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bursejan@gmail.com

Posts: 5,395
Registered: 9/25/16
Re: I rarely make silly mistakes, but Euler made a huge blunder in S
= Lim S

Posted: Oct 4, 2017 2:24 PM
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Bootstrapping the reals from the set Q, has
the advantage (or disadvantage), that the foundational
questions are split:

real
/ \
rational set/sequence (Dedekind/Cauchy)

Now in math, we probably don't run into the
https://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma

We simply go with:
The axiomatic argument, which rests on accepted precepts

So you can then dig further in each branch, you
might get possibly:

real
/ \
rational set/sequence (Dedekind/Cauchy)
| |
Peano ZFC

But this is not fixed, maybe you do rational
numbers group theoretic, who knows?

And possibly there is a glue somewhere on the
level of reals, some axioms also there?

Am Mittwoch, 4. Oktober 2017 20:17:05 UTC+2 schrieb burs...@gmail.com:
> 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.





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