ZFC supplies for the function Number of Unit Fractions between 0 and x the
result
NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
So, according to ZFC the number of unit fractions grows between 0 and (0,
1] by more than 2.
This is impossible because between any two unit fractions there are
"uncountably" many points which do not fit between 0 and (0, 1]. This
proves that ZFC does not deliver correct mathematics in this case, like in
several others:
https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory

Regards, WM

WM formulated the question :
> ZFC supplies for the function Number of Unit Fractions between 0 and x the
> result
> NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
> So, according to ZFC the number of unit fractions grows between 0 and (0, 1]
> by more than 2.
> This is impossible because between any two unit fractions there are
> "uncountably" many points which do not fit between 0 and (0, 1]. This proves
> that ZFC does not deliver correct mathematics in this case, like in several
> others:
> https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory
>
> Regards, WM

Yawn.

On Thursday, December 28, 2023 at 1:59:12 AM UTC-8, FromTheRafters wrote:
> WM formulated the question :
> > ZFC supplies for the function Number of Unit Fractions between 0 and x the
> > result
> > NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
> > So, according to ZFC the number of unit fractions grows between 0 and (0, 1]
> > by more than 2.
> > This is impossible because between any two unit fractions there are
> > "uncountably" many points which do not fit between 0 and (0, 1]. This proves
> > that ZFC does not deliver correct mathematics in this case, like in several
> > others:
> > https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory
> >
> > Regards, WM
> Yawn.

Beautiful facts of mathematics include that for ordinary set theory,
that Vitali's construction makes a non-measurable set, while, for
quasi-invariant measure theory, it results a doubling measure.

Similarly, the Banach-Tarski result has two quite very different
establishments, one of which is Hausdorff and Vitali's, geometrically.

These days there's a usual consideration for descriptive set theory,
that there's a combined track of sets and ordinals, and separate ones,
also.

I discuss this further at length in this stack of podcasts, or for example
some tens of thousands of writings to sci.math and sci.logic, which
altogether as a file fit in about a few minutes of full-motion video.

Philosophical Foreground, Analysis and Methods
https://www.youtube.com/playlist?list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY

Descriptive differential dynamics
https://www.youtube.com/playlist?list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5

Reading about Path Integral:
https://www.youtube.com/playlist?list=PLb7rLSBiE7F5xVz-L3jEuroDKA8hKLEGq

Reading from Maugin: Nonlinear Electromechanical Effects and Applications
https://www.youtube.com/playlist?list=PLb7rLSBiE7F6Dzc6mMXPfc4W9Y_OafJZj

Reading from Einstein's "Out of My Later Years"
https://www.youtube.com/playlist?list=PLb7rLSBiE7F41oobFHfUUar7iOwc5vNc3

On Thursday, December 28, 2023 at 10:08:35 AM UTC-8, Ross Finlayson wrote:
> On Thursday, December 28, 2023 at 1:59:12 AM UTC-8, FromTheRafters wrote:
> > WM formulated the question :
> > > ZFC supplies for the function Number of Unit Fractions between 0 and x the
> > > result
> > > NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
> > > So, according to ZFC the number of unit fractions grows between 0 and (0, 1]
> > > by more than 2.
> > > This is impossible because between any two unit fractions there are
> > > "uncountably" many points which do not fit between 0 and (0, 1]. This proves
> > > that ZFC does not deliver correct mathematics in this case, like in several
> > > others:
> > > https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory
> > >
> > > Regards, WM
> > Yawn.
> Beautiful facts of mathematics include that for ordinary set theory,
> that Vitali's construction makes a non-measurable set, while, for
> quasi-invariant measure theory, it results a doubling measure.
>
> Similarly, the Banach-Tarski result has two quite very different
> establishments, one of which is Hausdorff and Vitali's, geometrically.
>
> These days there's a usual consideration for descriptive set theory,
> that there's a combined track of sets and ordinals, and separate ones,
> also.
>
> I discuss this further at length in this stack of podcasts, or for example
> some tens of thousands of writings to sci.math and sci.logic, which
> altogether as a file fit in about a few minutes of full-motion video.
>
>
> Philosophical Foreground, Analysis and Methods
> https://www.youtube.com/playlist?list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY
>
> Descriptive differential dynamics
> https://www.youtube.com/playlist?list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5
>
> Reading about Path Integral:
> https://www.youtube.com/playlist?list=PLb7rLSBiE7F5xVz-L3jEuroDKA8hKLEGq
>
> Reading from Maugin: Nonlinear Electromechanical Effects and Applications
> https://www.youtube.com/playlist?list=PLb7rLSBiE7F6Dzc6mMXPfc4W9Y_OafJZj
>
> Reading from Einstein's "Out of My Later Years"
> https://www.youtube.com/playlist?list=PLb7rLSBiE7F41oobFHfUUar7iOwc5vNc3

https://www.youtube.com/watch?v=AhHPeYNLSzI&list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5&index=40

About 18:45, ...., about a quarter hour.

"Extra-ordinary", ....

....

On Thursday, December 28, 2023 at 11:05:25 AM UTC-8, Ross Finlayson wrote:
> On Thursday, December 28, 2023 at 10:08:35 AM UTC-8, Ross Finlayson wrote:
> > On Thursday, December 28, 2023 at 1:59:12 AM UTC-8, FromTheRafters wrote:
> > > WM formulated the question :
> > > > ZFC supplies for the function Number of Unit Fractions between 0 and x the
> > > > result
> > > > NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
> > > > So, according to ZFC the number of unit fractions grows between 0 and (0, 1]
> > > > by more than 2.
> > > > This is impossible because between any two unit fractions there are
> > > > "uncountably" many points which do not fit between 0 and (0, 1]. This proves
> > > > that ZFC does not deliver correct mathematics in this case, like in several
> > > > others:
> > > > https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory
> > > >
> > > > Regards, WM
> > > Yawn.
> > Beautiful facts of mathematics include that for ordinary set theory,
> > that Vitali's construction makes a non-measurable set, while, for
> > quasi-invariant measure theory, it results a doubling measure.
> >
> > Similarly, the Banach-Tarski result has two quite very different
> > establishments, one of which is Hausdorff and Vitali's, geometrically.
> >
> > These days there's a usual consideration for descriptive set theory,
> > that there's a combined track of sets and ordinals, and separate ones,
> > also.
> >
> > I discuss this further at length in this stack of podcasts, or for example
> > some tens of thousands of writings to sci.math and sci.logic, which
> > altogether as a file fit in about a few minutes of full-motion video.
> >
> >
> > Philosophical Foreground, Analysis and Methods
> > https://www.youtube.com/playlist?list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY
> >
> > Descriptive differential dynamics
> > https://www.youtube.com/playlist?list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5
> >
> > Reading about Path Integral:
> > https://www.youtube.com/playlist?list=PLb7rLSBiE7F5xVz-L3jEuroDKA8hKLEGq
> >
> > Reading from Maugin: Nonlinear Electromechanical Effects and Applications
> > https://www.youtube.com/playlist?list=PLb7rLSBiE7F6Dzc6mMXPfc4W9Y_OafJZj
> >
> > Reading from Einstein's "Out of My Later Years"
> > https://www.youtube.com/playlist?list=PLb7rLSBiE7F41oobFHfUUar7iOwc5vNc3
> https://www.youtube.com/watch?v=AhHPeYNLSzI&list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5&index=40
>
> About 18:45, ...., about a quarter hour.
>
>
> "Extra-ordinary", ....
>
>
> ...

Hodges' Muckenheim's hopeless is a pretty simple construct,
put up an insufferable, inflexible, incorrigeable idiot who makes
claims of a certain rhetorical position or in argument, thus that
any who otherwise start with similar principles are constantly
reminded that idiots are wrong, and so associated.

It's a trope.

If you haven't read "Hodges' hopeless", and study the development
of thinking about transfinite sets and the countable and uncountable,
you should, as it illustrates traps to avoid.

Also you can notice that my arguments and slates aren't included,
in Hodges', nor Mueckenheim's.

One should study Wertheimer about the learning of profound concepts.

https://en.wikipedia.org/wiki/Max_Wertheimer

https://en.wikipedia.org/wiki/Max_Wertheimer#Four_major_papers_from_1934_to_1940

Wertheimer sticks it to Karl Popper pretty good. "A Story of Three Days", .....

In at least one theory, the real Prof. Dr. WM has no idea Usenet exists at all,
or is quite confused about the whole thing. Luckily, in at least one theory
greater thinkers see quite all through and above it.

ZF supplies correct mathematics, ZFC including well-ordering with well-foundedness
includes more, then figuring it out as an exercise in the deconstructive,
for constructivism, and our canon and the conscientious.

....

Up next: more trolls, readily dispatched.

....

On Thursday, December 28, 2023 at 11:48:38 AM UTC-8, Ross Finlayson wrote:
> On Thursday, December 28, 2023 at 11:05:25 AM UTC-8, Ross Finlayson wrote:
> > On Thursday, December 28, 2023 at 10:08:35 AM UTC-8, Ross Finlayson wrote:
> > > On Thursday, December 28, 2023 at 1:59:12 AM UTC-8, FromTheRafters wrote:
> > > > WM formulated the question :
> > > > > ZFC supplies for the function Number of Unit Fractions between 0 and x the
> > > > > result
> > > > > NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
> > > > > So, according to ZFC the number of unit fractions grows between 0 and (0, 1]
> > > > > by more than 2.
> > > > > This is impossible because between any two unit fractions there are
> > > > > "uncountably" many points which do not fit between 0 and (0, 1]. This proves
> > > > > that ZFC does not deliver correct mathematics in this case, like in several
> > > > > others:
> > > > > https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory
> > > > >
> > > > > Regards, WM
> > > > Yawn.
> > > Beautiful facts of mathematics include that for ordinary set theory,
> > > that Vitali's construction makes a non-measurable set, while, for
> > > quasi-invariant measure theory, it results a doubling measure.
> > >
> > > Similarly, the Banach-Tarski result has two quite very different
> > > establishments, one of which is Hausdorff and Vitali's, geometrically..
> > >
> > > These days there's a usual consideration for descriptive set theory,
> > > that there's a combined track of sets and ordinals, and separate ones,
> > > also.
> > >
> > > I discuss this further at length in this stack of podcasts, or for example
> > > some tens of thousands of writings to sci.math and sci.logic, which
> > > altogether as a file fit in about a few minutes of full-motion video.
> > >
> > >
> > > Philosophical Foreground, Analysis and Methods
> > > https://www.youtube.com/playlist?list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY
> > >
> > > Descriptive differential dynamics
> > > https://www.youtube.com/playlist?list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5
> > >
> > > Reading about Path Integral:
> > > https://www.youtube.com/playlist?list=PLb7rLSBiE7F5xVz-L3jEuroDKA8hKLEGq
> > >
> > > Reading from Maugin: Nonlinear Electromechanical Effects and Applications
> > > https://www.youtube.com/playlist?list=PLb7rLSBiE7F6Dzc6mMXPfc4W9Y_OafJZj
> > >
> > > Reading from Einstein's "Out of My Later Years"
> > > https://www.youtube.com/playlist?list=PLb7rLSBiE7F41oobFHfUUar7iOwc5vNc3
> > https://www.youtube.com/watch?v=AhHPeYNLSzI&list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5&index=40
> >
> > About 18:45, ...., about a quarter hour.
> >
> >
> > "Extra-ordinary", ....
> >
> >
> > ...
> Hodges' Muckenheim's hopeless is a pretty simple construct,
> put up an insufferable, inflexible, incorrigeable idiot who makes
> claims of a certain rhetorical position or in argument, thus that
> any who otherwise start with similar principles are constantly
> reminded that idiots are wrong, and so associated.
>
> It's a trope.
>
> If you haven't read "Hodges' hopeless", and study the development
> of thinking about transfinite sets and the countable and uncountable,
> you should, as it illustrates traps to avoid.
>
> Also you can notice that my arguments and slates aren't included,
> in Hodges', nor Mueckenheim's.
>
> One should study Wertheimer about the learning of profound concepts.
>
> https://en.wikipedia.org/wiki/Max_Wertheimer
>
> https://en.wikipedia.org/wiki/Max_Wertheimer#Four_major_papers_from_1934_to_1940
>
> Wertheimer sticks it to Karl Popper pretty good. "A Story of Three Days", ....
>
> In at least one theory, the real Prof. Dr. WM has no idea Usenet exists at all,
> or is quite confused about the whole thing. Luckily, in at least one theory
> greater thinkers see quite all through and above it.
>
> ZF supplies correct mathematics, ZFC including well-ordering with well-foundedness
> includes more, then figuring it out as an exercise in the deconstructive,
> for constructivism, and our canon and the conscientious.
>
> ...
>
>
> Up next: more trolls, readily dispatched.
>
> ...

Mathesis is a word.

....