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Virgil
Dec 1, 2015, 5:29:37 PM12/1/15
to
Cantor's First Proof (of his theorem that the set of reals cannot be
enumerated) revisited and simplifed. It states that there cannot be any
surjection from the set of all naturals to the set of all reals).
Note: If one assumes every real has been indexed with a different
natural, it would otherwise follow below that some real must be indexed
by some natural larger than each of infinitely many other naturals.
Which is impossible!
Lemma: Assume every real has been indexed with a different natural.
Then every open real interval will contain two reals of lowest possible
index for that original open interval, and the interior points of this
new interval will necessarily have higher indices than its endpoints.
Proof of lemma: obvious!
Proof of theorem: Iterate the lemma to produce a nested sequence of such
closed intervals, each a subset of its predecessors and interior points
of each such interval having all indices greater than those of its own
endpoints and the endpoints of all prior such intervals.
Such an infinite sequence of nested closed real intervals is known to
have a non-empty intersection, at least one point interior to all those
intervals. But any such inside point must have a natural number index
larger than all the infinitely many different natural numbers indexing
different endpoints of the infinite sequence of intervals containing it.
Which is impossible in proper mathematics!
Thus any assumption that one can surject the naturals ONTO the reals is
proven false and the theorem is true, and the set of reals is
UNcountable!!!
--
Virgil
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)
Julio Di Egidio
Dec 1, 2015, 5:42:46 PM12/1/15
to
On Tuesday, December 1, 2015 at 10:29:37 PM UTC, Virgil wrote:
> Such an infinite sequence of nested closed real intervals is known to
> have a non-empty intersection,
Wrong in a first instance, then just false.
> Such an infinite sequence of nested closed real intervals is known to
> have a non-empty intersection,
Julio
Virgil
Dec 1, 2015, 5:59:07 PM12/1/15
to
In article <21ee5a2e-8460-4e94...@googlegroups.com>,
Julio Di Egidio <ju...@diegidio.name> wrote:
> On Tuesday, December 1, 2015 at 10:29:37 PM UTC, Virgil wrote:
>
> > Such an infinite sequence of nested closed real intervals is known to
> > have a non-empty intersection,
>
Julio is Wrong to dispute me and just stupidly false about the reals.
> On Tuesday, December 1, 2015 at 10:29:37 PM UTC, Virgil wrote:
>
> > Such an infinite sequence of nested closed real intervals is known to
> > have a non-empty intersection,
>
George Greene
Dec 1, 2015, 8:01:36 PM12/1/15
to
On Tuesday, December 1, 2015 at 5:42:46 PM UTC-5, Julio Di Egidio wrote:
> On Tuesday, December 1, 2015 at 10:29:37 PM UTC, Virgil wrote:
>
> > Such an infinite sequence of nested closed real intervals is known to
> > have a non-empty intersection,
>
> Wrong in a first instance
It is not wrong, idiot.
> On Tuesday, December 1, 2015 at 10:29:37 PM UTC, Virgil wrote:
>
> > Such an infinite sequence of nested closed real intervals is known to
> > have a non-empty intersection,
>
> Wrong in a first instance
Show somebody a sequence of nested closed real intervals that doesn't have a non-empty intersection, if you're so smart! Seriously, how can you say that and expect to be taken seriously??
George Greene
Dec 1, 2015, 8:03:18 PM12/1/15
to
On Tuesday, December 1, 2015 at 5:59:07 PM UTC-5, Virgil wrote:
> Julio is Wrong to dispute me and just stupidly false about the reals.
It's deeper than the reals. "Closed interval" is definable topologically in broader spaces than just the reals. It will be equally true IN THOSE that a sequence of nested closed intervals is going to have a non-empty intersection.
> Julio is Wrong to dispute me and just stupidly false about the reals.
I don't know why Julio thinks he can get away with it.
Julio Di Egidio
Dec 3, 2015, 6:50:59 AM12/3/15
to
On Wednesday, December 2, 2015 at 1:01:36 AM UTC, George Greene wrote:
> On Tuesday, December 1, 2015 at 5:42:46 PM UTC-5, Julio Di Egidio wrote:
> > On Tuesday, December 1, 2015 at 10:29:37 PM UTC, Virgil wrote:
> >
> > > Such an infinite sequence of nested closed real intervals is known to
> > > have a non-empty intersection,
> >
> > Wrong in a first instance
>
> It is not wrong, idiot.
You blind moron, as usual you completely miss the context and the point: it is *standard* mathematics that gets that wrong! Vases ending up empty in the limit, limit intersections but no limit indexes, and so on all long the line of just an incongruous approach to infinities of all kinds.
> On Tuesday, December 1, 2015 at 5:42:46 PM UTC-5, Julio Di Egidio wrote:
> > On Tuesday, December 1, 2015 at 10:29:37 PM UTC, Virgil wrote:
> >
> > > Such an infinite sequence of nested closed real intervals is known to
> > > have a non-empty intersection,
> >
> > Wrong in a first instance
>
> It is not wrong, idiot.
But here is a follow-up you might find more intelligible, which I did post already, though **in just 1 of** the 6 or 7 threads Virgil had quickly spammed with 10+ posts, here and in sci.math:
<quote>
[Virgil's] premise was: << If one assumes every real has been indexed with a different natural, it would otherwise follow below that some real must be indexed by some natural larger than each of infinitely many other naturals. Which is impossible! >>
No, the opposite is impossible, and rather incongruous is your mathematics!
Note: The intersection in question is *the limit of* a sequence of intersections. In fact, due to completeness, it must be degenerate, i.e. contain a single point: what is the natural index of that point? Omega, of course, i.e. the limit index. -- Of course there a limit index, i.e. just as much as there is a limit intersection.
</quote>
But you are an idiot, so you won't get it.
Julio
Virgil
Dec 3, 2015, 11:25:49 AM12/3/15
to
In article <3d512341-a609-445f...@googlegroups.com>,
But neither omega nor aleph_0 is a natural number, so there must be a
real point not having a natural number index!
Here is Cantor's first proof again:
Cantor's First Proof (of his theorem that the set of reals cannot be
enumerated) revisited and simplifed. It states that there cannot be any
surjection from the set of all naturals to the set of all reals).
Note: If one assumes every real has been indexed with a different
real point not having a natural number index!
>
> But you are an idiot, so you won't get it.
No! Julio Di Egidio is the idiot who won't let himself get it.
> But you are an idiot, so you won't get it.
Here is Cantor's first proof again:
Cantor's First Proof (of his theorem that the set of reals cannot be
enumerated) revisited and simplifed. It states that there cannot be any
surjection from the set of all naturals to the set of all reals).
natural, it would otherwise follow below that some real must be indexed
by some natural larger than each of infinitely many other naturals.
Which is impossible!
by some natural larger than each of infinitely many other naturals.
Which is impossible!
Lemma: Assume every real has been indexed with a different natural.
Then every open real interval will contain two reals of lowest possible
index for that original open interval, and the interior points of this
new interval will necessarily have higher indices than its endpoints.
Proof of lemma: obvious!
Proof of theorem: Iterate the lemma to produce a nested sequence of such
closed intervals, each a subset of its predecessors and interior points
of each such interval having all indices greater than those of its own
endpoints and the endpoints of all prior such intervals.
Then every open real interval will contain two reals of lowest possible
index for that original open interval, and the interior points of this
new interval will necessarily have higher indices than its endpoints.
Proof of lemma: obvious!
Proof of theorem: Iterate the lemma to produce a nested sequence of such
closed intervals, each a subset of its predecessors and interior points
of each such interval having all indices greater than those of its own
endpoints and the endpoints of all prior such intervals.
Such an infinite sequence of nested closed real intervals is known to
have a non-empty intersection, at least one point interior to all those
intervals. But any such inside point must have a natural number index
larger than all the infinitely many different natural numbers indexing
different endpoints of the infinite sequence of intervals containing it.
Which is impossible in proper mathematics!
Thus any assumption that one can surject the naturals ONTO the reals is
proven false and the theorem is true, and the set of reals is
UNcountable!!!
>
> Julio
intervals. But any such inside point must have a natural number index
larger than all the infinitely many different natural numbers indexing
different endpoints of the infinite sequence of intervals containing it.
Which is impossible in proper mathematics!
Thus any assumption that one can surject the naturals ONTO the reals is
proven false and the theorem is true, and the set of reals is
UNcountable!!!
>
Julio Di Egidio
Dec 3, 2015, 11:31:31 AM12/3/15
to
On Thursday, December 3, 2015 at 4:25:49 PM UTC, Virgil wrote:
> In article <3d512341-a609-445f...@googlegroups.com>,
> In article <3d512341-a609-445f...@googlegroups.com>,
Julio
Virgil
Dec 3, 2015, 12:55:40 PM12/3/15
to
In article <93f6490a-48c0-4c10...@googlegroups.com>,
> > Here is Cantor's first proof again:
Cantor's First Proof (of his theorem that the set of reals cannot be
enumerated) revisited and simplifed. It states that there cannot be any
surjection from the set of all naturals to the set of all reals).
Note: If one assumes every real has been indexed with a different
enumerated) revisited and simplifed. It states that there cannot be any
surjection from the set of all naturals to the set of all reals).
natural, it would otherwise follow below that some real must be indexed
by some natural larger than each of infinitely many other naturals.
Which is impossible!
by some natural larger than each of infinitely many other naturals.
Which is impossible!
Lemma: Assume every real has been indexed with a different natural.
Then every open real interval will contain two reals of lowest possible
index for that original open interval, and the interior points of this
new interval will necessarily have higher indices than its endpoints.
Proof of lemma: obvious!
Proof of theorem: Iterate the lemma to produce a nested sequence of such
closed intervals, each a subset of its predecessors and interior points
of each such interval having all indices greater than those of its own
endpoints and the endpoints of all prior such intervals.
Then every open real interval will contain two reals of lowest possible
index for that original open interval, and the interior points of this
new interval will necessarily have higher indices than its endpoints.
Proof of lemma: obvious!
Proof of theorem: Iterate the lemma to produce a nested sequence of such
closed intervals, each a subset of its predecessors and interior points
of each such interval having all indices greater than those of its own
endpoints and the endpoints of all prior such intervals.
Such an infinite sequence of nested closed real intervals is known to
have a non-empty intersection, at least one point interior to all those
intervals. But any such inside point must have a natural number index
larger than all the infinitely many different natural numbers indexing
different endpoints of the infinite sequence of intervals containing it.
Which is impossible in proper mathematics!
Thus any assumption that one can surject the naturals ONTO the reals is
proven false and the theorem is true, and the set of reals is
UNcountable!!!
>
intervals. But any such inside point must have a natural number index
larger than all the infinitely many different natural numbers indexing
different endpoints of the infinite sequence of intervals containing it.
Which is impossible in proper mathematics!
Thus any assumption that one can surject the naturals ONTO the reals is
proven false and the theorem is true, and the set of reals is
UNcountable!!!
>
> Sure, you just keep spamming...
It is a proof valid in mathematics even if not in JDE's wee world!
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