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A Counter-argument against Lebesgue Measure Theory?

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Iosephus Granicae

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Sep 2, 2021, 10:08:34 PM9/2/21
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I occasionally saw a webpage claiming that there's inherent paradox inside Lebesgue Measure Theory, where he constructs a perfect set with positive measure and being nowhere dense, while he claims that the set cannot have positive measure, because it consists of "denumerable" "singular isolated points", hence the paradox occurs.

And also he claims he solved that "paradox" with "limiting condition", but he hasn't provide any sound and unambiguous definition about that concept.
Maybe someone with similar opinion can contact him.

The links are here:
www.jamesrmeyer.com/infinite/lebesgue-measure.html
And:
www.jamesrmeyer.com/infinite/understand-infinity-and-limits.html

Gus Gassmann

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Sep 3, 2021, 8:00:32 AM9/3/21
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The argument is nonsense, and it falls apart right at the beginning.

Every interval I -- "complete" or not -- of A has *rational* endpoints, namely n/m +/- 10^(-k) for suitable n, m and k. That the endpoints of I have further intervals constructed around them (which intersect I and are therefore *not* complete) is immaterial.

I did not try to make sense of the rest. In my opinion, the author is a crank, and I have no inclination to pursue this further.

William

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Sep 3, 2021, 12:30:38 PM9/3/21
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The author makes many mistakes. He does not ensure that the endpoints are irrational, and his definition of a "complete interval" is faulty.
However, these errors can be corrected. The basic problem is that he only considers limits of endpoints to be "between the intervals".
And indeed this set is countable. He does not consider limits of limits, or limit of limits of limits, and so forth, including "limits of limits of limits of ..."

Essentially the same argument was presented by WM.

--
William Hughes

Mike Terry

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Sep 3, 2021, 2:52:31 PM9/3/21
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On 03/09/2021 03:08, Iosephus Granicae wrote:
> I occasionally saw a webpage claiming that there's inherent paradox inside Lebesgue Measure Theory, where he constructs a perfect set with positive measure and being nowhere dense, while he claims that the set cannot have positive measure, because it consists of "denumerable" "singular isolated points", hence the paradox occurs.

This is rather strange - his argument /exactly/ mirrors one of WM's
arguments, which has been discussed on sci.math multiple times. And he
has exactly the same "blind spot" mistake WM always makes! Both WM and
JMR claim that the set only contains points which are end points of the
"complete intervals" of the complement set. That is simply not the
case, as can be demonstrated by constructing simple counter examples,
although I doubt it would be possible to get WM or JMR to accept those.
So the claim that the set is denumerable doesn't hold water - neither WM
nor JMR offer any actual /proof/, just wishy-washy wordy arguments and
unsupported claims. I'd even suspect WM and JMR were the same person,
except that both have independent records on the internet, so I don't
think that's the case. Probably

The web page is littered with examples of crank language, which is a bit
of a give-away:
- ...according to /conventional mathematics/..
- ...Some people when faced with this /unpalatable contradiction/ make
the /bizarre attempt/...
- ...somehow (although exactly how is /never divulged/)...
- ...welcome to /fantasy land/...
- ...their /beloved/ Lebesgue Theory...
[italics added by me]

and so on. Practically every other line screams CRANK. :)

>
> And also he claims he solved that "paradox" with "limiting condition", but he hasn't provide any sound and unambiguous definition about that concept.
> Maybe someone with similar opinion can contact him.

If you want to point out his errors, why don't /you/ email him? But I
have a feeling it would be like water off a duck's back.

Regards,
Mike.

markus...@gmail.com

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Sep 3, 2021, 3:47:55 PM9/3/21
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Without looking, it just seems just a proof that the set is not measurable.

Benson Bear

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Feb 28, 2023, 12:44:36 PM2/28/23
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On Friday, September 3, 2021 at 2:52:31 PM UTC-4, Mike Terry wrote:
> This is rather strange - his argument /exactly/ mirrors one of WM's
> arguments, which has been discussed on sci.math multiple times. And he
> has exactly the same "blind spot" mistake WM always makes! Both WM and
> JMR claim that the set only contains points which are end points of the
> "complete intervals" of the complement set. That is simply not the
> case, as can be demonstrated by constructing simple counter examples,

Coming to this late after starting an exchange with this guy.

I have been doing just what you say (check the most recent
comments on JMR Lebesgue Measure page). I have whittled
it down to a very simple example and somehow he still does
not get it. I am baffled but still think we can get him to see
this. Or, if he does not (and refuses to) then make his
assumptions very clear (e.g. perhaps he will say I cannot refer
to an infinite sequence of intervals, which seem strange
because he does so himself).

Can you give a reference to the WM discussion of this particular
argument?

Mike Terry

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Mar 1, 2023, 12:33:34 PM3/1/23
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Here are a few (all from sci.logic):

The first time it came up, I think:

sci.logic 29/5/12 Subject: Matheology § 022
https://groups.google.com/g/sci.logic/c/b2txw8tfFzU/m/no9zlOFnQGgJ


Then it came up again several years later:

sci.logic 22/11/18 Covering all rational points by irrational intervals
msgid: <0259e7d8-951c-4e52...@googlegroups.com>
https://groups.google.com/g/sci.logic/c/0Dzy1OjA0Aw/m/rI0lCm4FAwAJ

and [this was a biggie thread!]

sci.logic 31/12/18 Clusters and Cantor dust
msgid: <df87465e-8633-4ba0...@googlegroups.com>
https://groups.google.com/g/sci.logic/c/5A6TuGC5UCg/m/l3v6K9bcFQAJ


WM has had a few aliases he has used over the years: If you see WM, ganzhintersehen, transfinity*
are all WM.

Regards,
Mike.

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