Re: How to re-establish mathematics?

[WM]

It is rather simple to re-establish mathematics. Whenever you talk about real numbers or large sets add the phrase: "each element being definable by a finite string over a countable alphabet".

In mathematics that is redundant but not much disturbing and it's definitely worth the trouble!

Regards, WM

[Python]

Doktor "Venom" Frankenheim wrote:
> It is rather simple to re-establish mathematics. Whenever you talk about real numbers or large sets add the phrase: "definable by a finite string over a countable alphabet".

>
> In mathematics that is redundant but not much disturbing and it's definitely worth the trouble!

Mathematics are very well established, Herr Mueckenheim. What need
to be re-established, nevertheless, is intellectual decency at
Hochschule Augsburg, what need to be removed there is the absolute
scandal of hosting a crank as a teacher for years, abusing students
by his usurpated autority, same for any position you had or have in
German Academy.

[Dan Christensen]

On Tuesday, May 29, 2018 at 7:39:35 AM UTC-4, WM wrote:
> It is rather simple to re-establish mathematics. Whenever you talk about real numbers or large sets add the phrase: "each element being definable by a finite string over a countable alphabet".
>
> In mathematics that is redundant but not much disturbing and it's definitely worth the trouble!
>

Thanks for confirming that this idiocy of yours in NOT currently a feature of modern mathematics.


Dan

[WM]

Am Dienstag, 29. Mai 2018 13:42:22 UTC+2 schrieb Python:
> Doktor "Venom" Frankenheim wrote:
> > It is rather simple to re-establish mathematics. Whenever you talk about real numbers or large sets add the phrase: "definable by a finite string over a countable alphabet".
> >
> > In mathematics that is redundant but not much disturbing and it's definitely worth the trouble!
>
> Mathematics are very well established, Herr Mueckenheim. What need
> to be re-established, nevertheless, is intellectual decency at
> Hochschule Augsburg,

So you take undefinable objects in mathematics as guaranteeing intellectual decency? Let me guess, your brain damage must be at least 20 % with no positive prognosis for recovery. (To confirm that by yourself simply ask some students who have not yet been corrupted by matheology.)

Regards, WM

[Ross A. Finlayson]

A "re-establishment" of mathematics
would just be another wing in
"Hilbert's Infinite Living Musuem".

Then, the closer that it's correct,
it's also a working model of the place.


Stare into the void, see.

[Python]

Doktor 'venom' Frankenheim wroet:

> Am Dienstag, 29. Mai 2018 13:42:22 UTC+2 schrieb Python:
>> Doktor "Venom" Frankenheim wrote:
>>> It is rather simple to re-establish mathematics. Whenever you talk about real numbers or large sets add the phrase: "definable by a finite string over a countable alphabet".
>>>
>>> In mathematics that is redundant but not much disturbing and it's definitely worth the trouble!
>>
>> Mathematics are very well established, Herr Mueckenheim. What need
>> to be re-established, nevertheless, is intellectual decency at
>> Hochschule Augsburg,

>> , what need to be removed there is the absolute
>> scandal of hosting a crank as a teacher for years, abusing students
>> by his usurpated autority, same for any position you had or have in
>> German Academy.

> So you take undefinable objects in mathematics as guaranteeing intellectual decency? Let me guess, your brain damage must be at least 20 % with no positive prognosis for recovery. (To confirm that by yourself simply ask some students who have not yet been corrupted by matheology.)

I take you as a venom and a shame, a crackpot and a liar, a dishonest
person and a abusive teacher. I take YOU for what YOU are.

Never use the word 'decency' in front of anyone, you are a venom.

[WM]

Am Mittwoch, 30. Mai 2018 03:38:07 UTC+2 schrieb Python:

> >>> It is rather simple to re-establish mathematics. Whenever you talk about real numbers or large sets add the phrase: "definable by a finite string over a countable alphabet".
> >>>
> >>> In mathematics that is redundant but not much disturbing and it's definitely worth the trouble!
> >>
> >> Mathematics are very well established, Herr Mueckenheim. What need
> >> to be re-established, nevertheless, is intellectual decency at
> >> Hochschule Augsburg,
> >> , what need to be removed there is the absolute
> >> scandal of hosting a crank as a teacher for years, abusing students
> >> by his usurpated autority, same for any position you had or have in
> >> German Academy.
>
> > So you take undefinable objects in mathematics as guaranteeing intellectual decency? Let me guess, your brain damage must be at least 20 % with no positive prognosis for recovery. (To confirm that by yourself simply ask some students who have not yet been corrupted by matheology.)
>
> I take you as a venom and a shame, a crackpot and a liar, a dishonest
> person and a abusive teacher.

That is the foremost symptome of your intellectual illness. I advised you to confirm that decent mathematicians use only defined element by simply asking some students who have not yet been corrupted by matheology or other sane people in your surrouding, if available. But of course you refuse that. Another symptom of your insanity. I hope you are at least equipped with a sane body.

By the way, even some set theorist are decent enough: "if the ZFC axioms of set theory are consistent, then there are models of ZFC in which every object, including every real number, every function on the reals, every set of reals, every topological space, every ordinal and so on, is uniquely definable without parameters." [J.D. Hamkins et al.: "Pointwise definable models of set theory", arXiv (2012)]

Regards, WM

[Zelos Malum]

Why should we "re-establish" it when it is well established alraedy?

[WM]

Am Donnerstag, 31. Mai 2018 08:11:50 UTC+2 schrieb Zelos Malum:
> Den tisdag 29 maj 2018 kl. 13:39:35 UTC+2 skrev WM:
> > It is rather simple to re-establish mathematics. Whenever you talk about real numbers or large sets add the phrase: "each element being definable by a finite string over a countable alphabet".
> >
> > In mathematics that is redundant but not much disturbing and it's definitely worth the trouble!

> Why should we "re-establish" it when it is well established alraedy?

What you call *math* is a perverted *mess* of contra-mathematical delusions. In mathematics everything that can be applied has to be defined.

Regards, WM

[Zelos Malum]

>
What you call *math* is a perverted *mess* of contra-mathematical delusions. In mathematics everything that can be applied has to be defined.

Says you and you are completely wrong.

No where in the rules of logic does it say that we need to write a finite string of symbols to define it for it to exist. no NOL says it.

[Dan Christensen]

On Tuesday, May 29, 2018 at 7:39:35 AM UTC-4, WM wrote:

> It is rather simple to re-establish mathematics. Whenever you talk about real numbers or large sets add the phrase: "each element being definable by a finite string over a countable alphabet".
>

It seems you just can't get over the fact that real analysis with its uncountable set of real numbers has been wildly successful as the basis for modern science and technology. Unless you are able to achieve all of the same results more easily, your goofy system will be a non-starter with scientists and engineers. I'm afraid they will insist on having real numbers like pi and root 2. And not just approximations, but actual numbers.

After proving 1=/=2 in your goofy system (hee, hee!), you should probably establish your "simplified" version of the Intermediate Value Theorem. https://en.wikipedia.org/wiki/Intermediate_value_theorem

Then your "simplified" version the Fundamental Theorem of Calculus would be nice. https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

Then you should be well on your way "reestablishing" mathematics as we know it. So, get busy and quit wasting your time (and ours) here. You have your work cut out for you.

Destiny calls. Seize the day, Mucke!


Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[WM]

Am Montag, 4. Juni 2018 08:04:44 UTC+2 schrieb Zelos Malum:
> >

> No where in the rules of logic does it say that we need to write a finite string of symbols to define it for it to exist. no NOL says it.

Mathematics is based upon this simple truth: What cannot be defined as an object of mathematics is not an object of mathematics.

Matheology has other principles. They lead to ridiculous assertions like: the complete infinite Binary Tree has more paths than nodes. Obviously not all paths can be distinguished by nodes. Application to architecture yields the surprising recognition, there are more houses than bricks.

Regards, WM

Regards, WM

[Zelos Malum]

>Mathematics is based upon this simple truth: What cannot be defined as an object of mathematics is not an object of mathematics.

Says who? You?

You are aware that this risks getting into the situation where you prove something must exist AND is impossible to write as a string of symbols, ergo resulting in a contradiction, right?

>Matheology has other principles. They lead to ridiculous assertions like: the complete infinite Binary Tree has more paths than nodes. Obviously not all paths can be distinguished by nodes. Application to architecture yields the surprising recognition, there are more houses than bricks.

The principle is Ex(P(x)) evaluates to true, it runs in no risk to get into a contradiction and it is more universal and has nothign arbitrary to it.

[WM]

Am Montag, 4. Juni 2018 15:07:09 UTC+2 schrieb Zelos Malum:

It is rather simple to re-establish mathematics. Whenever you talk about real numbers or large sets add the phrase: "each element being definable by a finite string over a countable alphabet".

> You are aware that this risks getting into the situation where you prove something must exist AND is impossible to write as a string of symbols, ergo resulting in a contradiction, right?

Wrong. If you get into that case, then you can describe the way there and the restrictions imposed. That is defining the unkown object by "a finite string over a countable alphabet".

Regards, WM

[Ross A. Finlayson]

"Real analysis", yes, "uncountable", no.

There aren't any "applications" courtesy "uncountable":
because the real numbers' analytical character is
courtesy "countable additivity".

How to "re-establish" mathematics is the same as the
way to establish it. Trans-finite cardinals have a
place in mathematics, but for example as there's a
universe, the foundation (to establish) is otherwise.

[Dan Christensen]

On Tuesday, June 5, 2018 at 11:58:17 AM UTC-4, Ross A. Finlayson wrote:
> On Monday, June 4, 2018 at 4:16:28 AM UTC-7, Dan Christensen wrote:
> > On Tuesday, May 29, 2018 at 7:39:35 AM UTC-4, WM wrote:
> > > It is rather simple to re-establish mathematics. Whenever you talk about real numbers or large sets add the phrase: "each element being definable by a finite string over a countable alphabet".
> > >
> >
> > It seems you just can't get over the fact that real analysis with its uncountable set of real numbers has been wildly successful as the basis for modern science and technology. Unless you are able to achieve all of the same results more easily, your goofy system will be a non-starter with scientists and engineers. I'm afraid they will insist on having real numbers like pi and root 2. And not just approximations, but actual numbers.
> >
> > After proving 1=/=2 in your goofy system (hee, hee!), you should probably establish your "simplified" version of the Intermediate Value Theorem. https://en.wikipedia.org/wiki/Intermediate_value_theorem
> >
> > Then your "simplified" version the Fundamental Theorem of Calculus would be nice. https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
> >
> > Then you should be well on your way "reestablishing" mathematics as we know it. So, get busy and quit wasting your time (and ours) here. You have your work cut out for you.
> >
> > Destiny calls. Seize the day, Mucke!
> >
> >
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com
>
> "Real analysis", yes, "uncountable", no.
>

Good luck working out the details.

> There aren't any "applications" courtesy "uncountable":
> because the real numbers' analytical character is
> courtesy "countable additivity".
>

If you want irrational numbers like pi or root 2 (i.e. numbers with non-repeating decimal expansions), you will need uncountably many real numbers. From Cantor, we know that it is impossible even in theory to construct a countably infinite list of all the decimal expansions of the numbers between 0 and 1.

[Ross A. Finlayson]

Thanks, Dan, and good luck to you.

Indeed my studies include a countable model of
the real numbers, where of course we're familiar
with and trust the complete ordered field as is
modern mathematics' abstract algebra's usual
development.

This is where "continuity" is a strong property
and is evinced with line-drawing (or the sweep,
the "equivalency function", as you may well know
I maintain is building a countable continuous
domain).

With a continuous domain there's built the familiar
IVT and FTC's etc then to leave the rest to
real analysis.

This is a different, complementary model of the
reals, for line continuity first then for the
field continuity as above with which we are all
familiar.

There are also proofs of the ir-rationality of
for example e or pi constructively, as it were,
and it's been established since the Pythagoreans
that there are ir-rational numbers as among all
the rational numbers as classical constructions
provide.

It's also well-established that both the rationals
and ir-rationals are each: everywhere dense, nowhere
continuous, and whose complement is each other in
the reals.

This then is for a "signal continuity" that (as
for example built in the bounded and extended to
the unbounded) that "doubling the density" of the
rationals is a "continuous signal".

[Dan Christensen]

How about the Intermediate Value Theorem?

[Ross A. Finlayson]

Courtesy the naturals [0,oo) shrunk down
to the interval [0,1].

Or, f(n) = n/d, n -> d, d -> oo .

[Julio Di Egidio]

On Tuesday, 5 June 2018 18:20:02 UTC+2, Dan Christensen wrote:
> On Tuesday, June 5, 2018 at 11:58:17 AM UTC-4, Ross A. Finlayson wrote:

<snip>

> > "Real analysis", yes, "uncountable", no.
>
> Good luck working out the details.

Yet constructive analysis exists (and it's not even too complicated): but you
are too stupidly dishonest not to keep acting as if all but the interior of
your ass is just alien...

Julio

[burs...@gmail.com]

Fruit cake at its best.

{ L_n | L_n = lim d->oo n/d } = {0}

This aint working. Do you know what you
are doing, or are you just stupid?

[burs...@gmail.com]

If you want to map [0,oo) to [0,1) you
could use for example:

f(n) = n/(n+1)

You can also map the extended reals
from [0,oo] to [0,1], if you would set:

f(oo) = 1

Because

lim n->oo f(n) = 1.

[Julio Di Egidio]

On Tuesday, 5 June 2018 19:24:38 UTC+2, burs...@gmail.com wrote:
> Am Dienstag, 5. Juni 2018 19:16:19 UTC+2 schrieb Ross A. Finlayson:
> > Courtesy the naturals [0,oo) shrunk down
> > to the interval [0,1].
> >
> > Or, f(n) = n/d, n -> d, d -> oo .
>
> Fruit cake at its best.
>
> { L_n | L_n = lim d->oo n/d } = {0}
>
> This aint working. Do you know what you
> are doing, or are you just stupid?

But that's not what he wrote, you other utter moron and resident spammer.

*Plonk*

Julio

[burs...@gmail.com]

Who cares what you wrote Chulio?

*Plonk*

[Ross A. Finlayson]

Host: Oh God.

Elk: All brontosauruses are thin at one end, much MUCH thicker in the middle, and then thin again at the far end. That is the theory that I have and which is mine, and what it is too.

Host: That's it, is it?

Elk: Right, Chris.

Host: Well, Anne, this theory of yours seems to have hit the nail on the head.

Elk: And it's mine.

Host: (ironical) Thank you for coming along to the studio.

[Julio Di Egidio]

On Tuesday, 5 June 2018 19:30:25 UTC+2, burs...@gmail.com wrote:

> Who cares what you wrote Chulio?

Meaning what?? That you truly are a fucking spammer??

Who the fuck PAYS for all this time you can waste, never mind the fucking
damage?!

Fuck you, Jan Burse, and whomever pays your bills.

*Plonk*

Julio

[burs...@gmail.com]

FC (Fruit Cake), with some goodwill your
notation could be interpreted as the
rational numbers,

like just all n/d, but there is no IVT
in the rational numbers. What if your
root is irrational?

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

Am Dienstag, 5. Juni 2018 19:24:38 UTC+2 schrieb burs...@gmail.com:

[burs...@gmail.com]

There is no courtesy towards Dans question.
The naturals [0,oo) shrunk down to the
interval [0,1], is impossible solution

if you want to have all the irrational
numbers in the range, since there are
uncountable many irrationals,

there are of course some mappings from
N to some constructive irrationals. But
there are uncountable many continous

functions, so I guess there are also
uncountable many IVT roots. You would
need to do the following for your quest:
a) Exhibit your constructive irrational numbers
b) Exhibit your constructive function space
c) And show that it still has IVT

One known such exhibit is the a) algebraic numbers
and b) polynomials with algebraic number coefficients.
Probably some form c) of IVT can be established:

See also:

Can every polynomial with algebraic coeffitients
be transformed into a polymonial with integer
coefficients and at least some of the same roots?
https://math.stackexchange.com/q/1430084/4414

[Dan Christensen]

On Tuesday, June 5, 2018 at 1:16:19 PM UTC-4, Ross A. Finlayson wrote:
> On Tuesday, June 5, 2018 at 9:43:39 AM UTC-7, Dan Christensen wrote:

> >
> >
> > How about the Intermediate Value Theorem?
> >

>

> Courtesy the naturals [0,oo) shrunk down
> to the interval [0,1].
>
> Or, f(n) = n/d, n -> d, d -> oo .

Huh??? Here's how it works:

Let f be a defined continuous function on [a, b] and let s be a number with f(a) < s < f(b). Then there exists at least one x in [a, b] with f(x) = s.
https://en.wikipedia.org/wiki/Intermediate_value_theorem

Since it looks like you are working with constructive proofs, you will have to construct of example of x given continuous function f. Can you do it?

[Dan Christensen]

On Tuesday, June 5, 2018 at 1:20:04 PM UTC-4, Julio "Plonk Man" Di Egidio wrote:
> On Tuesday, 5 June 2018 18:20:02 UTC+2, Dan Christensen wrote:
> > On Tuesday, June 5, 2018 at 11:58:17 AM UTC-4, Ross A. Finlayson wrote:
> <snip>
> > > "Real analysis", yes, "uncountable", no.
> >
> > Good luck working out the details.
>

> Yet constructive analysis exists...


Indeed, any theorem in constructive analysis is a theorem in classical analysis. The converse doesn't appear to be true, however. Can scientists and engineers really afford to do without all these missing tools?

[Ross A. Finlayson]

It's ran(f) for which these properties
exist, here incognizant of the complete
ordered field usually written R. For
this I refined the notation as about
R^\bar and R^\dots and R^\hat.

There is that f(n) = x is in [0,1], and,
f^{-1}(x) is some incredibly large number,
basically effectively infinite, but

f(a) < f(b) => a < b

and

f(a) < s < f(b) => a < f^{-1}(s) < b .

It's basically worked up that there does
not exist 0 < s < 1 for which s isn't in ran(f).

There's much light on this from "Correct presentation
of the diagonal argument" as was written on sci.logic.

So, ran(f) is a countable continuous domain [0,1]
establishing the IVT and thus quite usually the FTC's.

[Ross A. Finlayson]

Ah, you're mistaken, I am not "fruit cake".

About fruitcake:

"I love fruitcake it's delicious,
I miss it fondly and particularly
because I loved fruitcake,
it's not the point to go out and get
one, I know I love fruitcake already.

Even though I miss it, I'll love fruitcake
long after it's gone, not for any other,
but the ideal of fruitcake lasts forever."


Now, if you would, consider further these
properties of ran(f). ("Sweep", the
"Equivalency Function".) You can find that
for the antidiagonal, nested intervals, and
mw-proof (or "Cantor's real first"), it's
unique among functions N -> R[0,1] that the
uncountability doesn't hold. (And here
that R[0,1] is ran(f) and it's established
as a continuous domain.)

[burs...@gmail.com]

Hey fruit cake, the word "continous" is a property
of a function, not of a domain. For domain there are
other words, with different meaning.

The IVT needs a value space and a function space.
Usually the value space are the reals R, and
the function spaces is f : R -> R, f continoues.

So this here:

> So, ran(f) is a countable continuous domain [0,1]
> establishing the IVT and thus quite usually the FTC's.

Where you use your strange f to construct a value
space, is completely useless. You need also
construct a function space.

If you have a countable infinite value space,
[0,1], how do you even a continous function
f : [0,1] -> [0,1] ? Why should anything you

wrote **establish** IVT. You are halucinating and
confused. You even don't understand what the objects
of the IVT are. You just write nonsense as usual.

[burs...@gmail.com]

You can define continuous f : A -> A, even if A
is countable infinite.(*) But the IVT doesn't
hold necessarily anymore, because you need to

pick a zero from A. But the IVT exactly says
that such an element from A always exists.
So the problem is not the domain, but have

a function space with roots. The classical IVT
is not constructive concerning the roots,
so it cannot be the same function space,

if you are looking for something constructive.
But your sweep and other fruit cake lingo,
anyway make no sense. What do you want to archive?

Show us a constructive function space, where
there are examples, like the algebraic polynomials,
or do you want to show us your inner clowno?

(*)
Sorry for not providing you a reference, but
for example for a continuous function on R, I
guess you only need the values on Q.

[Dan Christensen]

On Tuesday, June 5, 2018 at 5:32:09 PM UTC-4, Ross A. Finlayson wrote:
> On Tuesday, June 5, 2018 at 1:07:31 PM UTC-7, Dan Christensen wrote:
> > On Tuesday, June 5, 2018 at 1:16:19 PM UTC-4, Ross A. Finlayson wrote:
> > > On Tuesday, June 5, 2018 at 9:43:39 AM UTC-7, Dan Christensen wrote:
> >
> > > >
> > > >
> > > > How about the Intermediate Value Theorem?
> > > >
> >
> > >
> > > Courtesy the naturals [0,oo) shrunk down
> > > to the interval [0,1].
> > >
> > > Or, f(n) = n/d, n -> d, d -> oo .
> >
> >
> > Huh??? Here's how it works:
> >
> > Let f be a defined continuous function on [a, b] and let s be a number with f(a) < s < f(b). Then there exists at least one x in [a, b] with f(x) = s.
> > https://en.wikipedia.org/wiki/Intermediate_value_theorem
> >
> > Since it looks like you are working with constructive proofs, you will have to construct of example of x given continuous function f. Can you do it?
> >
> >

>

> It's ran(f) for which these properties
> exist, here incognizant of the complete
> ordered field usually written R. For
> this I refined the notation as about
> R^\bar and R^\dots and R^\hat.
>
> There is that f(n) = x is in [0,1], and,
> f^{-1}(x) is some incredibly large number,
> basically effectively infinite, but
>

How do you define f^{-1}? You have to construct it (i.e. prove its existence).

> f(a) < f(b) => a < b
>
> and
>
> f(a) < s < f(b) => a < f^{-1}(s) < b .
>
> It's basically worked up that there does
> not exist 0 < s < 1 for which s isn't in ran(f).
>
> There's much light on this from "Correct presentation
> of the diagonal argument" as was written on sci.logic.
>
> So, ran(f) is a countable continuous domain [0,1]
> establishing the IVT and thus quite usually the FTC's.

So, how do you construct x as given in the statement of IVT (above)???

Maybe start with an example. Suppose we have continuous f: [0,2] --> R such that f(x)=x^2 and f(0)=0 and f(2)=4. Construct x such that x in [0, 2] and f(x)=2.

[Ross A. Finlayson]

The reciprocal is shown to exist,
not what it is,
except preserving the ordering
of both domain and range.

Clearly this EF is "not a real function",
with examples like Dirac impulse or
Heaviside step, built from real functions,
modeled by them.

This has f^{-1} exists in N, the domain,
not what it is except for f^{-1}(0) = 0.

https://math.stackexchange.com/questions/283192/what-are-the-features-of-n-d-n-rightarrow-d-d-rightarrow-infty-n-d-in

[Ross A. Finlayson]

We're familiar with continuity as about the
Dedekind-completeness, courtesy Least Upper
Bound (LUB, a usual definition) or gaplessness.

That's "field continuity", where usual definitions
of field continuity are equivalent.

This setting sees a "line continuity". Basically
each element f(n) is the "limit" of the sequence
(f(0), ..., f(n-1)).

There's also a notional "signal continuity" as
where the rationals are dense, and if they were
twice as dense, that would reconstruct the real line.

[burs...@gmail.com]

How many continuities do you have fruit cake?
Any of them related to some IVT, or all of
them just some arbitrary word salad nonsense?

[Ross A. Finlayson]

That's three different definitions, Burse.

Or maybe you only know the one?

[burs...@gmail.com]

How many IVTs do you have? Also 3? But there
is only one value domain? So the same domain A,
you have

C1 = { f : A -> A | continuity 1 }

C2 = { f : A -> A | continuity 2 }

C3 = { f : A -> A | continuity 3 }

Can you enlighten us, which one you intend to
use for which kind of IVT? Or do you just like
to produce some intelligent sounding text,

as usual on sci.math?

[burs...@gmail.com]

Corr: Typo,

C1 = { f : A -> A | f is continuous_1 }

C2 = { f : A -> A | f is continuous_2 }

C3 = { f : A -> A | f is continuous_3 }

[Ross A. Finlayson]

"Continuous domains".

[Ross A. Finlayson]

I'll write more later,
but the more you search
my posts for these things
the more you'll find.

(And they maintain a consistent
theme and development.)

Here clearly the point is to
have a neat self-contained synopsis
here about "a countable model of
a continuous domain of [0,1]" and
then furthermore in terms of
theory generally.

[wpih...@gmail.com]

On Tuesday, June 5, 2018 at 1:20:02 PM UTC-3, Dan Christensen wrote:
>
> If you want irrational numbers like pi or root 2 (i.e. numbers with non-repeating decimal expansions), you will need uncountably many real numbers.

Piffle. There are lots of irrational computable numbers. (e.g. the two you mentioned).

[wpih...@gmail.com]

On Monday, June 4, 2018 at 10:07:09 AM UTC-3, Zelos Malum wrote:

> You are aware that this risks getting into the situation where you prove something must exist AND is impossible to write as a string of symbols

No, if you take the positions that only computable objects exist, and that showing that (P doesn't exist) is untrue does not imply the existence of P,
(to the extent that it is sensible to say that WM takes a position, these are positions WM takes) you are not exposing yourself to any risk.

--
William Hughes

[Dan Christensen]

Oh, really??? So no IVT in your system. Good luck selling that nonsense to scientists and engineers. I think you find they are very attached to irrational numbers like root 2 -- not just approximations + or -, but an actual number.

They already have a system that consistently works in real-world applications. Why should they abandon it for yours?

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[Dan Christensen]

We are still waiting for your proof of the Intermediate Value Theorem for continuous functions on whatever alternative you may have to real numbers.

[burs...@gmail.com]

See you later aligator. So your nonsense is
competely baseless as usual. Just word salad.

[Ross A. Finlayson]

https://math.stackexchange.com/questions/1596022/difference-between-real-functions-and-real-valued-functions

It's not a matter of
line continuity xor field continuity:
it's a matter of
line continuity and field continuity.

Interesting properties of this
first bridge between the discrete and continuous
include for example that it's a CDF and "a"
(surprisingly enough not "the") pdf of the
natural integers at uniform random,
surprising because otherwise we know that
probability distributions are defined by a
unique pdf.

So, this example (and you'll notice counterexample
to uncountability and otherwise no bridge between
discrete and continuous) has other usual unique
properties as among our combined constructions and
here in reference to otherwise the uniqueness up
to isomorphism of the complete ordered field and
otherwise the sufficiency of Dedekind-completeness.

(Or "Eudoxus/Dedekind/Cauchy is insufficient", not
so much for what it shows, but for what it doesn't
show here in terms of the bridge between
discrete and continuous.)

So, then why scientists and engineers might find
any extra utility in this result that what there
is of the analytical character of the reals as
courtesy countable additivity, then is about the
aspects of points in or on a line then in higher
dimensions, and about how points are two-sided on
the line then 3/4/5 sided on the plane, etc, then
about for example Banach-Tarski, Vitali, and, the
re-Vitali-ization of measure theory.

Establishing the IVT (for the FTC's) is one thing,
here there's also _more_ about the numbers from
this approach, where uncountability becomes
expressly mute about what would contradict itself
in terms of the perfect results of the analytical
character of the reals and countable additivity.

There's a quote from the mathematical appendices to
R. Penrose's "Fashion, Faith, and Fantasy in the
New Physics of the Universe" (2016) that I wanted
to share with you.

"Cantor's theory of (cardinal) infinities is really
concerned just with _sets_, which are not thought of
as being structured as some kind of continuous space.
For our purposes here, we do need to take into
account continuity (or smoothness) aspects of the
spaces that we are concerned with."

This is following:

"I mention Cantor's theory here only so as to
make the contrast with what we are doing here,
which is different."

[Ross A. Finlayson]

I have put forward
quite a very simple construction,
here's an interesting extended
article from Terence Tao in his
paradigm of hard and soft analysis
about the resort of analysts to
topologists' ultrafilters and
ultraproducts to build over set
theory ideas of bridges between
discrete and continuous, and,
establishment of the properties
of continuous spaces.

https://terrytao.wordpress.com/2013/12/07/ultraproducts-as-a-bridge-between-discrete-and-continuous-analysis/

[Ross A. Finlayson]

In this context it seems reasonable
to survey results for "bridge between
discrete and continuous".

Eg https://www.google.com/search?q=bridge+between+discrete+and+continuous

Maybe this will help you understand
better the motivation and direction
of finding these features in the foundation,
then of course for a formalist's course
in the simple establishment of mathematical
fundamentals about them.

[Dan Christensen]

Ummmm... thanks for the link, I guess, but this STILL does not establish your equivalent of the all-important Intermediate Value Theorem for whatever set or other structure you have in mind to replace the uncountable set of real numbers.

[Ross A. Finlayson]

The point is to establish a parallel or analog
to Least Upper Bound, that each limit of the
elements of the range is in the range.

The idea here is that this "convergence" for
a limit is defined for f(n) as the elements
(f(0), ..., f(n-1)). Unlike the familiar Cauchy
criterion and about the equivalence classes
of sequences that are Cauchy, it's about a
rather-reduced* Cauchy criterion and that the
equivalence classes of the sequences x_n
= (f(0), ..., f(n-1)) are singletons.

(Basically there's only one representation
and one for each and they are only all
together, the values in this space [0,1]
that is ran(f).)

So, each of these sequences satisfies this
convergence* criterion, only one each and
one for each.

This way LUB* is built then "measure 1.0"*
from f(d) - f(0) = 1, so to reflect the shared
result of the IVT.

So, there's quite a bit the _less_ machinery
then for suitable reduced definitions of
convergence and limit to establish that
there aren't any elements in ran(f) that
aren't limits of convergent* sequences in
ran(f).

Here the asterisks are basically to mark
these notions of parallels or equivalences
to the usual notions, of convergence and
limit, in this "discrete" setting.



And, again, this isn't so much "replacement"
as "augmentation", of the means to establish
continuous domains.

Of course it's necessary that this "Equivalency
Function" fall out of the number-theoretic
results otherwise establishing uncountability
of continuous domains without contradiction.
Via inspection you'll note it does.

[Ross A. Finlayson]

This is from the "constant monotone strictly
increasing" where the constant is infinitesimal.
(And the increase is bounded, in ran(f), for
where it is unbounded in N.)

[burs...@gmail.com]

What else? You sure you didn't forget something
in your word salad. You already listed infinites-
simal, measurement, etc.. etc..

[Ross A. Finlayson]

Here are some links via a sci.math/sci.logic
portal to some various posts with more about
what we have here as talking points.

https://groups.google.com/d/msg/sci.logic/J6hOI2iQKxg/X6qeCzjXCgAJ
https://groups.google.com/d/msg/sci.logic/C1l6ihQI62k/I5CEsvGJcj4J
https://groups.google.com/d/msg/sci.logic/23hPLaiLsQY/GVD9isdt2z8J
https://groups.google.com/d/msg/sci.logic/MwmhihH_lTU/CcUUu-cPBAAJ

There's quite the few thousand more.

[burs...@gmail.com]

Funny word salada. Here is an example from
fruit cakes first link:

"countable model of R[0,1]" (eg as countable
additivity about measure 1.0).

I dont think a sigma algebra is a countable.
What is the usual homework here?

Re: Prove that cardinality of an sigma-algebra
is finite or uncountable
https://math.stackexchange.com/q/320035/4414

[burs...@gmail.com]

Maybe you have problems with counting,
1, omega, 2^omega, ... Ha Ha

[Ross A. Finlayson]

That's "about" measure 1.0.

Measure theory along with Least Upper Bound
are tacked on definitions to "set theory"
and its models of number systems, for real
analysis.

I think we can agree on that, though some have
the LUB isn't stipulated, but for these people
Dedekind cuts are rationals.

So, the idea of this neater, smaller, simpler,
more fundamental establishment of a continuous
domain (which I think we can agree is a
connected space) is to standup what
"measure 1.0" provides for real analysis.

This function has properties from a
self-contained development that are
ignored or prohibited in the later
usual curriculum's complete ordered field.
These are relevant to the establishment
then of identical features in the resulting
structure as so identify a background for
continuum analysis, historically known as
infinitesimal analysis and these days as
real analysis.

Then, for a setting later in measure theory,
is to involve the countable additivity of
the usual development of the analytical
character of "the" (or, "a") model of the
real numbers, geometry's number line.


So "about measure 1.0" is _about the role_
of measure 1.0 because the means or mechanism
as so entails its significance.

[Ross A. Finlayson]

One idea about this ran(f) and measure is
that f(n) is the the value and also the measure,
and it ranges zero to one, with that the only
measurable sets are as of n-sets (initial segments)
of N.

"... For this reason, one considers instead
a smaller collection of privileged subsets
of X. These subsets will be called the
measurable sets. They are closed under
operations that one would expect for
measurable sets, that is, the complement
of a measurable set is a measurable set and
the countable union of measurable sets is a
measurable set. Non-empty collections of
sets with these properties are called σ-algebras."
-- https://en.wikipedia.org/wiki/Sigma-algebra#Measure

That the complement of one of these sets as
measurable would also be measurable is
basically that the complement of the (unique)
set with measure x is 1-x and that f(n) = x is
of N \ {0, ..., n}.

Here you could note we'd be working with the
inverses of f for x: n = f^-1(x) 0 < x < 1 without
giving values for n except it has those values.

Does it not meet the definition?

https://en.wikipedia.org/wiki/Sigma-algebra#Definition_and_properties

[Ross A. Finlayson]

Or, here the measurable sets could be those
for 0 <= a < b <= 1 that of natural integers
{ m = f^{-1}(a), ..., n = f^{-1}(b) }, basically this
has here that there are countable measurable
sets (or "the countable sigma-algebra") courtesy
that the domain is countable.

No?

[Dan Christensen]

Yes, the completeness property of the real numbers. https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers

Oh, but we are not talking about the real numbers, are we? Some countable domain, right? Exactly what countable domain ARE you talking about? The rational numbers? We know that they are NOT complete.

[Ross A. Finlayson]

No, this is only "infinity" shrunk down to "one"
preserving the order and relative distance and
only in order and only all together.

On a side note the rationals are irrationals
are both dense in the reals, and each "dense"
in each other in the sense as you will recall
that between any two standard ir-rationals
there are infinitely many rationals and
vice versa, all nested together.

Then for "how many of those are there" when
their topological properties are the same,
gets into "well-ordering the reals".

"Well-ordering the reals" is another example
of an existence result, without usually
actually providing one, where this here
appears to be an example.

[Dan Christensen]

On Wednesday, June 6, 2018 at 8:10:00 PM UTC-4, Ross A. Finlayson wrote:
> On Wednesday, June 6, 2018 at 5:02:49 PM UTC-7, Dan Christensen wrote:

> > > > Ummmm... thanks for the link, I guess, but this STILL does not establish your equivalent of the all-important Intermediate Value Theorem for whatever set or other structure you have in mind to replace the uncountable set of real numbers.
> > > >
> > > >

> > > The point is to establish a parallel or analog
> > > to Least Upper Bound, that each limit of the
> > > elements of the range is in the range.
> > >
> >
> > Yes, the completeness property of the real numbers. https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers
> >
> > Oh, but we are not talking about the real numbers, are we? Some countable domain, right? Exactly what countable domain ARE you talking about? The rational numbers? We know that they are NOT complete.
> >
> >

> No, this is only "infinity" shrunk down to "one"
> preserving the order and relative distance and
> only in order and only all together.
>
> On a side note the rationals are irrationals
> are both dense in the reals, and each "dense"
> in each other in the sense as you will recall
> that between any two standard ir-rationals
> there are infinitely many rationals and
> vice versa, all nested together.
>
> Then for "how many of those are there" when
> their topological properties are the same,
> gets into "well-ordering the reals".
>
> "Well-ordering the reals" is another example
> of an existence result, without usually
> actually providing one, where this here
> appears to be an example.

So, the domain is NOT the set of rational numbers? If so, how would you define the domain in question?

[Zelos Malum]

>Wrong. If you get into that case, then you can describe the way there and the restrictions imposed. That is defining the unkown object by "a finite string over a countable alphabet".

But we proved that it cannot be written as such, a contradiction.

[WM]

Am Donnerstag, 7. Juni 2018 07:43:00 UTC+2 schrieb Zelos Malum:
> >Wrong. If you get into that case, then you can describe the way there and the restrictions imposed. That is defining the unkown object by "a finite string over a countable alphabet".
>
> But we proved that it cannot be written as such, a contradiction.

Give a concrete example.

Regards, WM

[burs...@gmail.com]

If the total recursive functions were listable,
the Goldbach Conjecture would already have an
answer yes or no.

Thats just an idea. Didn't try it yet to translate
this problem:

Goldbach Conjecture is true/false

To this problem:

function XYZ is total yes/no

Obviously this function here:

function XYZ() {
the first even x such that
there are not two primes p,q
with x=p+q
}

Would return a value if Goldbach Conjecture
were false. The check x such that
two primes p,q with x=p+q is primitive

recursive. But then finding such an x
when Goldbach Conjecture would be true
is impossible

[Ross A. Finlayson]

We're talking about ran(f), "sweep",
"the Equivalency Function", where
f(n) = n / d, n -> d, d-> oo
that has domain N and range < [0,1],
with f(0) = 0 and f(d) = f(oo) = 1.

The domain, dom(f) is N, the range,
ran(f), is claimed to be a "continuous
domain".

Here a "continuous domain" is about
establishing that ran(f) is gapless,
i.e. that a chart of ran(f) (in order)
is the same as drawing a line segment.

This eventually results as "Cantor
proves the line is drawn" and for
example reflects a geometry underneath
Euclid's with points and spaces instead
of points and lines.

[burs...@gmail.com]

So you following the steps of John Garbagiel,
establishing yourself as a cranky cranky crank?

Ha Ha, Cantor proves the line is drawn. Which
proof does do this? Can you tell us?

Some fruit cake word salad proof?

[Dan Christensen]

Good luck establishing or reestablishing anything at all with that notion. Maybe I just don't have "the vision thing," but it sounds like a REALLY tough sell to me.

[Ross A. Finlayson]

Uncountability has that the line can't
be built by its recursive divisions,
of the complete ordered field. (There's
modern mathematics' abstract algebras'
complete ordered field of each of its
elements, established with LUB that it's
Dedekind-complete then given analytical
character with further defining measure 1.0.)

This particular counterexample EF has
that the line is "drawn" as from infinitely
many points only and directly and smoothly
between zero and one.

So, noting "Cantor proves the line is (only) drawn"
has that with this function as unique counterexample
to uncountability, that the range or co-image as it
were meets the topological and geometrical properties
of a line. The line as this image isn't formed from
dots on the line, but always drawing from zero to
one, in a countable (countably infinite) manner.

This is a neat logical development.

I think we can agree that the putrid JG-bot or
"infinite foul toot" equivocated the tangent
line and rise/run as a false definition of
derivative, and good riddance. Or, "retro-
finitist crankety trolls can step off."

So, no, I've maintained that "EF is
(unique) counterexample to uncountability"
here in terms of what is found as another
model of the real numbers, the smooth
line segment, basically since I heard of
uncountability, for a fundamental bridge
between the discrete, and the continuous.

That was really the goal of my research,
to understand the bridge and divide
between the discrete and continuous.

Here I found it took quite a bit the
less machinery to establish it in these
"line-drawing" terms, alongside a formal
education with a usual mathematics curriculum,
to be then putting them together.

Basically Cantor proves the line isn't not drawn.

Otherwise we're left to the result of what
would be "well-ordering the reals" to so
draw it.

So, well-order the reals.

[Ross A. Finlayson]

An underlying geometry aside, and it's a matter
of axiomatics and basically fundamental analytic
geometry with Euclid as a platform above with
provided definitions of points and lines, I
think we can agree that it's not _easy_ to
have enough research in the frontiers of today's
researches in modern mathematics, to find a
post-modern retro-classical approach as
reconciles them together.

It rather requires quite strong adherence to
both, or each of the, strong properties as
of properties of the continuum, here for
technical meanings for points "in" (a line)
and "on" (a given line).

Goedel for incompleteness of the ordinary
development does have that there are more
properties of the self-same objects (as
of their elements or here our model), so,
it's not like modern mathematics disqualifies
itself from further development.

Then finding regular/ordinary set theory as
a platform above sets more generally (the
extra-ordinary or non-well-founded), is about
then how underlying assumptions in the quite
totally primitive are sussed apart to have a
theory with all the mathematical (constant,
consistent, complete, for a platonist concrete)
objects in it, besides those of the platform
of today's fundamental theorems of the field
of real analysis.

So, it's not easy to have everything true
(everything that is true, everything only true)
in the theory, but, it's rather the demand of
a universal theory (for example a theory with
a universe) that it does.

[burs...@gmail.com]

"The line as this image isn't formed from
dots on the line, but always drawing from zero to
one, in a countable (countably infinite) manner."

And where does Cantor say that? Except
that you fruit cake say that? Why theorem
aka proof of Cantor implies that?

[burs...@gmail.com]

Maybe we should make a sanity test with fruit
cake word salad. Here is the question: Does
every subset of rational numbers have

a minimal element? Does every "line" as some
subset of Q, with further properties not yet
specified, but for sure finitely many or countable

infinite many elements, have a left end?

[Ross A. Finlayson]

Wouldn't that be, an, insanity test?

"Elements of the σ-algebra are called measurable sets. ...."


".... So, well-order the reals."

[burs...@gmail.com]

Already for like 10 years, you recycle and
copy here the same word salad. Do you have
anything to say? I don't think so.

It sounds all like this here:

The Continuum is Countable: Infinity is Unique
Laurent Germain - September 2008

"In this paper, I show that there is a bijection
between the set of integers and the set of real n
umbers. I show that the set R is countable and that
there is only one dimension for infinite sets, @."
https://arxiv.org/pdf/0809.4144.pdf

His baseless proof is very stunning:

Proposition 2.3
Proof: Note that the tree representing the set
N, which at each node links up with 10 branches,
includes the infinite subtree with 2 branches
that represents P(N).

Ha Ha

[burs...@gmail.com]

Mostlikely the problem is, that you
can get really high from baseless proofs.
Its like math meth:

This is how the paper closes, sounds very
familiar fruit cakes solution:

In this article, I prove that the cardinality of
infinite sets is always @. There is a unique dimension
for infinity. I also prove that infinity is always
countable. The consequence of this result is that
the continuum is a countable set. This result has
several consequences in Mathematics, Probability
and Statistics. It modifies not only our vision of
the world, but also that of modeling in Physics,
Economics, Biology and Computer Science, among other
fields. Moreover, it opens the door to new
concepts in Philosophy."

[Ross A. Finlayson]

Would you expect something different?

An interesting feature of the
infinite balanced binary tree is
that the depth-first traversal is
the breadth-first traversal.

The analog of Cantor's result to the
infinite balanced binary tree sees
a counterexample in the sweep of the
leaves.

Otherwise you can catalog Germain's
under Hodges' hopeless.

[Ross A. Finlayson]

It's rather underdefined, where here
instead you can note there's discussed
various different definitions of continuity.

All of the same linear continuum, eg the
long line of du Bois-Reymond, about there
being crossings of the real line of the space
of functions from reals to reals, which you'd
admit is rather large.

For "infinite sets are equivalent", this is
for example about a universal set that would
be its own powerset. You should be familiar
with Cantor's paradox and how his domain
principle (that the universe exists) was
formalized out by Zermelo and Fraenkel who
neatly axiomatized infinity as ordinary, in
the milieu of Russell and of Frege and about
how the combinatorics of the finite are laid
aside the aspect of the infinite yet still
that they're never to meet.

Maybe you should look back in the attic
about Kunen inconsistency (a reflection
on Cantor's paradox) and then try to
reconcile views as from Spinoza for an
integer continuum as for "the" continuum
(of numbers).

These are philosophers, yes.

Trans-finite cardinals' contribution to
real analysis ends at countable additivity.

[burs...@gmail.com]

You cannot traverse an infinite tree depth-first.

[burs...@gmail.com]

I guess fruit cake has absolutely no
clue what an infinite tree means.

Here is a finite trees:

*
/ \
0 1
/ \ / \
00 01 10 11

And here are two traversals:

depth-first, *, 0, 00, 01, 1, 10, 11

breadth-first, *, 0, 1, 00, 01, 10, 11

The requirement is a natural number indexing.

0 1 2 3 4 5 6

Now take an infinite and balance tree. depth-first
will only enumerate one branch and go astray,
never come back.

[Ross A. Finlayson]

The leaves: sweep the leaves.

[burs...@gmail.com]

What leaves, breadth-first and depth-first
are not leave algorithms.

https://en.wikipedia.org/wiki/Breadth-first_search

https://en.wikipedia.org/wiki/Depth-first_search

Could it be, that you are completely out of
your mind? Like john gabriel, birdy birdy

brainy just right now?

[burs...@gmail.com]

The leave enumeration, for a finite tree,
is the same for both:

And here are two traversals:

depth-first, *, 0, 00, 01, 1, 10, 11

breadth-first, *, 0, 1, 00, 01, 10, 11

If you only look at the leaves, you get:

depth-first, 00, 01, 10, 11

breadth-first, 00, 01, 10, 11

But this holds only for finite trees. For
completely infinite trees, means each branch
is infinite, both algorithms

will not deliver any leaves. An completely
infinite tree doesn't have any terminal
leaves. Thats the definition of infinite tree.

But a breadth-first algorithm can enumerate
all nodes of an infinite tree if finite branching,
whereas a depth-first algorithm cannot enumerate

all nodes of an infinite tree with finite branching.

[burs...@gmail.com]

For a finite tree, the breadth first enumeration
might give a different leave enumeration, a

different order, if the branch have not all
the same depth. Like for example this tree:

*
/ \
0 1
/ \

00 01

breadth first will have this leaf enumeration 1, 00, 01,
depth first will have this leaf enumeration 00, 01, 1
but for an infinite tree, there is no sweep via

depth first of the nodes. Thats just complete nonsense.
And there is no sweep, I take here sweep as traversal,
for a leaf nodes, in an infinite tree at all. Never,

because an infinite tree doesn't have leaf nodes.
If it had leaf nodes, it wouldn't be infinite.

[Ross A. Finlayson]

I rather enjoy Jan Burse (or his usual
incarnation, as it were, of his prolog-
founded sockpuppet troll), just to say
so that it's refreshing to enjoy his
usual insights even if there isn't
maintained much of a decorum.

So, that said, what is of interest here
is as above (after we've taken away this
thread from WM): that these measurable sets
of elements from ran(f) form a sigma-algebra.

It'd be unlike JB to leave out a point he
could call out as a fault, here I'd wonder
his impression of what this neat simple
structure's definition sees follow as a
direct interface to what is otherwise a
slightly cumbersome adaptor between the
dispersion of the uncountable reals to
result with only the countable additivity
as the nature of the reals with respect to
real analysis.

No?

[Ross A. Finlayson]

Here basically the idea is that
if it did, they would be having
the same values as ran(f) and in
the same order and otherwise as
that all the resulting detail
could so result in a linear system.

This is again where via inspection
this function (not a real function,
but a special function as it were,
among many various families of
special functions) doesn't see
uncountability result for
otherwise functions N <-> [0,1]
as would be "onto".

[burs...@gmail.com]

You are halucinating and confused as usual.
There are no sockpuppets involved.

Why should somebody need sockpuppets to tell
you that you are totally out of your

mind, and subject to a phantasm orgy of
math gibberish? Nothing what you

write makes any sense at all.

[burs...@gmail.com]

Even why are you so complicated? Why talk
about a function f : N -> [0,1].
Normal people would simply say a subset

A of the interval [0,1] that is countable.
This subset could automatically have the
real ordering? Why do you need your function f?

Thats realy annoying. Because in the IVT, we
also talk about functions, but these are
of course other functions.

But you keep posting miles of stuff here,
only to say a ran(f) is your domain? A
subset A of [0,1] is also a domain, and

if this subset A is countable then there
will exist a function f : N -> A, which
is onto. At least onto A. Europeans would

say surjective. But again. Nobody cares
about such a function unless it has some
special properties or you can write it down.

But so far you are probably only interested
in a countable subset A from [0,1], which
in your opinion can solve the IVT.

Unfortunately the IVT doesn't deal with a
subset A from [0,1], so you again what your
homework would be:

- define your IVT

- consisting of your domain, where
you will take the roots from, and
also the whole range of the functions.

- consisting of your continous functions
space inside the IVT, for which you want
to show there are roots.

You even struggle in defining a domain.
What do you want to sweep leafs for, in an
infinite tree, if there are no leafs,

this will not give you your domain.

[burs...@gmail.com]

Corr.:

- consisting of your domain, where
you will take the roots from, and

also the domain for each of the functions.

[burs...@gmail.com]

You don't get more or less circular or
impredicative if you would directly start
talking of your domain A, instead indirectly

of some ran(f). Even if you explain it
as a countable subset of [0,1], this is not
necessarely circular or impredicative, or

presupposes the existence of the uncountable
[0,1]. Just take it as a motivation preamble,
the [0,1] will drop out, if you will solely

talk about your domain A, and show us how
your IVT based on this domain A looks like,
and why it should hold.

You see no [0,1], which you avoid, needed.
You would only need to show that your A is
countable, and that it has some

properties of a line. I don't see what it
buys you to use:

A = ran(f)

Also your d -> oo, is a little strange,
and often hangs in the clouds, you don't
write consequently f_d then.

you could of course also talk about
domain families:

A_1, A_2, ..., A_d, ...

Maybe your A and all of your (A_d) are
related. Who knows? In my opinion:

L_n = lim d->oo n/d = 0

So far your notation doesn't give anything,
and needs a lot of improvement.

Am Freitag, 8. Juni 2018 01:37:12 UTC+2 schrieb burs...@gmail.com:

[Ross A. Finlayson]

Thanks / MfG.

The properties of the real functions
that model the special function
establish for the range, as a set:

a) "0" is always contained
b) "1" is always contained
c) the f(n)'s are:
constantly,
strictly monotonically,
increasing.

This is just to say that the
range is a "subset" of [0,1],
without having yet defined
of what "set" [0,1] "is", or
how it is modeled.

It's established that there's
no neighborhood in [0,1] not
containing an element of ran(f).

Because we know Cantorian set theory,
via inspection this function is not
eliminated from being a bijection because
the number-theoretic results for uncountability
don't apply. (This could be left off for
a constructive proof but it's eventually
reconciled and not a contradiction.)

Then, there's this "reduced" notion of
convergence, that each (f(0), ... f(n-1))
"converges" to f(n). In this model that's
the only sequence that so does, there are
only finite (unbounded) sequences of elements
with limits in this model. This is to be shown
the same as LUB.

This way then the IVT is established.

"Measure 1.0" is so built as above with length
directly or in terms of a neat sigma-algebra as
above.

I'm glad you can at least see the outline,
(or an outline) where I don't just accept
but appreciate your comment and demands
for formalism.

For reading off n/d as numerator/denominator,
it does have as about infinity, then, yes.

[burs...@gmail.com]

Is your answer even english. I don't understand
a word. What language is this supposed to be. What
should "eliminated from being a bijection" mean?

And why do you write "0" and "1" with apostroph.
I still think you are plain crazy. What you
present is a big mess.

Its the same mess as WM does when he defines
a set |E, but you cannot define a set |E
by a disjuntion like |E={2} v |E={2,4} v ...

Some with you. How many functions do you have?
Something with f=f1 v f=f2 v... and then
ran(f) doesn't give you a domain.

You are still high on math meth and only
produce gibberish. You are worse than WM in
dealing with mathematical objects.

Am Freitag, 8. Juni 2018 02:37:09 UTC+2 schrieb Ross A. Finlayson:

[Ross A. Finlayson]

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

[Ross A. Finlayson]

The impulse function is an example of a
generalized function or as I learned the
term "function modelled by real functions".

That's just to illustrate that not only
can a function be built this way, and that
they are, also that not all mathematicians
are or were aware of these means.

Why I wrote "0" and "1" is because in the
formulation starting with the natural integers,
there hasn't necessarily been built anything
like the rationals or complete ordered field.
So, these are 0 and 1. Properties of values
existing between zero and one would be
established in usual inequalities. This is
that, given the trichotomy of values in the
integers (here just the natural integers)
that n > m => f(n) > f(m), strictly. It doesn't
say the values are in abstract algebra's usual
rational ordered field or complete ordered field,
it just has them ranging from zero to one.

The next point is about how these values are
equi-distributed as it were in what we know of
as each neighborhood in [0,1]. This is where
there are points in the neighborhood of 0 and
points in the neighborhood of 1 and points in
the neighborhoods of all the points in-between.

Then this notion of gaplessness is basically
defining convergence as of the elements from
the beginning through each (of the unbounded
integers). This castrated convergence criterion
that is bounded for each, can be seen the same
as convergence usually, with filling in the
bounded sequences with zero difference for
all the following terms. It doesn't say for
a given integer n what the critical point (obs.)
or limit point or limit is, just that it exists
and for each neighborhood in [0,1]. Also it
has that the gaplessness is courtesy all the
limit points are in the range.

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

Then the IVT is as from the LUB.

Then as 1 - 0 = b - a = 1.0 for measure 1.0,
that ticks all the boxes for a very simple
establishment of the fundamental theorems of
calculus about [0,1].


Burse is normally so poised, I can't quite
imagine what's so shaken him.

[Zelos Malum]

Den torsdag 7 juni 2018 kl. 15:23:41 UTC+2 skrev WM:
> Am Donnerstag, 7. Juni 2018 07:43:00 UTC+2 schrieb Zelos Malum:
> > >Wrong. If you get into that case, then you can describe the way there and the restrictions imposed. That is defining the unkown object by "a finite string over a countable alphabet".
> >
> > But we proved that it cannot be written as such, a contradiction.
>
> Give a concrete example.
>
> Regards, WM

using ZFC we can easily prove that there exist things is not definable in finite strings.

[burs...@gmail.com]

Is your answer even english. I don't understand
a word. What language is this supposed to be. What

should "Then the IVT is as from the LUB" mean?

[mlwo...@wp.pl]

And using marksism-leninism you can easily prove that
communism is the best.