Virgil
Posts:
4,661
Registered:
1/6/11
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Re: On the diagonal argument again
Posted:
May 17, 2012 6:32 PM
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In article <jp3t0p$3pr$1@speranza.aioe.org>,
"LudovicoVan" <julio@diegidio.name> wrote:
> "Virgil" <virgil@ligriv.com> wrote in message
> news:virgil-53D379.16004217052012@bignews.usenetmonster.com...
> > In article <jp3quu$u4s$2@speranza.aioe.org>,
> > "LudovicoVan" <julio@diegidio.name> wrote:
> >> "Virgil" <virgil@ligriv.com> wrote in message
> >> news:virgil-A21B1A.14021217052012@bignews.usenetmonster.com...
> >> > In article <jp2hbk$5er$2@speranza.aioe.org>,
> >> > "LudovicoVan" <julio@diegidio.name> wrote:
> >> >> "Virgil" <virgil@ligriv.com> wrote in message
> >> >> news:virgil-EE5F05.20194016052012@bignews.usenetmonster.com...
> >> >>
> >> >> > You may consider as you like, but that doesn't make it so.
> >> >>
> >> >> You rather have (yet again) no right to spam my threads. Get lost
> >> >> Virgil,
> >> >> you and your *zero content*: just fuck off my way.
> >> >
> >> > Sci.math is an open Newsgroup.
> >> >
> >> > When you post to sci.math, anyone reading those posts has as much right
> >> > to post comments on them as you have to post them in the first place.
> >>
> >> WRONG!! You have no fucking right of breaking all etiquettes, of usenet,
> >> of
> >> civil life, and eventually of law. That I cannot literally kick you in
> >> the
> >> ass out of that door doesn't confer you any bloody right at all. FUCK
> >> OFF
> >> Virgil, liar, moron and spammer, GO SPAM SOMEWHERE ELSE.
> >
> > What alleged law do you claim I am breaking by publicly pointing out
> > your asininity?
>
> You are concrete damage with your spam: thanks to you and the others that
> follow, threads get to hundreds of posts of nonsense, lies and ad hominem,
> and this is an actual damage that makes any true exchange impossible.
>
> So fuck off spam somewhere else, Virgin moron and liar.
Those who meet polite discussion with unwarranted rudeness and name
calling are the only ones causing any damage to this discussion.
My own proof of Cantor's theorem:
*********************************************
A PROOF, after Cantor's first proof, OF THE UNCOUNTABILITY OF THE REALS
I.e. although inclusion of |N in |R as a subset gives an injection from
|N to |R,
so that Card(|N) <= Card(|R), the theorem states that
there is no surjection from |N to |R, so that Card(|N) < Card(|R)
THEOREM: There cannot be a surjection from |N to |R.
The present proof is based on the difference between the order
properties of the standard real line and the order properties of the set
of naturals numbers or, equivalently, the real line.
Namely:
A strictly increasing sequence of naturals CANNOT have a natural as its
limit.
A strictly increasing but bounded sequence of reals MUST have a real
limit, its least upper bound, and that limit must be strictly greater
than each and every member of the sequence.
And similarly for strictly decreasing bounded sequences of reals.
PROOF of the theorem:
If there were a surjection from |N to |R then each real would be paired
with a natural so that different reals are paired with different
naturals and vice versa and with none of either unpaired.
If we assume this is possible and has been done,
we will show that this leads to a contradiction, namely existence of a
natural larger than infinitely many other naturals.
Assume a bijection between |N and |R exists and has been implemented
Take the two reals with the lowest naturals as endpoints of a real
interval.
It is clear that all the interior points of this real interval must be
paired with naturals larger that those of its endpoints.
Now take the two reals interior to that interval with the lowest paired
naturals to be the endpoints of a subinterval of that interval.
Since the naturals are naturally well ordered, this must be possible.
It is clear that the interior points of this real interval also must be
paired with naturals larger that those of the endpoints.
By repeating this process ad infinitum one generates from each such
interval a proper subinterval, giving an infinite, nested, decreasing,
but never empty, sequence of closed real intervals.
Each such interval contains only points with higher attached naturals
than those of its endpoints and contains infinitely many such points.
The properties of the real number system requires that the intersection
of such a nested sequence of closed intervals is not empty, and the
alleged bijection between |N and |R requires the natural associated
with any of its members is necessarily larger than each of the
infinitely many natural numbers associated with endpoints those
infinitely many nested intervals which contain it.
But there is no natural number larger than infinitely many other natural
numbers, which is impossible. Thus there is AT LEAST one real number in
that intersection, and possibly many of them, in that non empty
intersection of nested intervals, for which there is no possible
corresponding natural.
This is a contradiction which can only have been caused by our original
assumption that the There could be a surjection from |N to |R, and thus
can only be resolved by rejecting the possibility of any surjection from
the set of naturals to the set of reals.
QED!
While I have never seen this particular form of argument in the
literature of countability and Cantor's theorem, it is so clear and
incontrovertible that I doubt that I am the first to have found it.
*********************************************
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