§ 336

WM

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Sep 8, 2013, 5:36:31 AM9/8/13

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§ 336

Annix's question "Is the analysis as taught in universities in fact the analysis of definable numbers?"
(cp. § 332) has raised another comment by Andrej Bauer:

This is off-topic, but: it makes no sense to claim that "constructivist continuum is countable in ZFC sense". What might be the case is that there is a model of constructive mathematics in ZFC such that the continuum is interpreted by a countable set. {{Why then not use this as the model of university mathematics and drop all blather about uncountability?}} Indeed, we can find such a model, but we can also find a model in which this is not the case. {{this is no contradiction, of course.}} Moreover, any model of ZFC is a model of constructive set theory {{and constructive models contraditct uncountability because everything constructed, say as a constructed anti-diagonal of a constructed Cantor-list, is well-defined and therefore definable and therefore belonging to a countable set}} You see, constructive mathematics is more general than classical mathematics, and so in particular anything that is constructively valid is also classically valid {{for instance the theorem that uncountability does never occur constructively}}.

Regards, WM

Virgil

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Oct 5, 2013, 8:32:27 PM10/5/13

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In article <2bae9a2c-2637-4081...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> § 336
>
> Annix's question "Is the analysis as taught in universities in fact the
> analysis of definable numbers?"

It is the analysis of the definable real number system, which
necessarily has more numbers in it than WM can define.
>
> Regards, WM
--

WM

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Oct 12, 2013, 9:33:02 AM10/12/13

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On Sunday, 6 October 2013 02:32:27 UTC+2, Virgil wrote:
> It is the analysis of the definable real number system, which necessarily has more numbers in it than WM can define.

If you want to well-order a set you have to index every element, such that every subset has a first element. How do you index uncountably many elements with only countably many numbers/indices?

Regards, WM

fom

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Oct 12, 2013, 10:34:12 AM10/12/13

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Just about the same way in which you place the
nonconstructive limit of a collection of constructive
representations into that collection.

Your contradiction arises by imagining something
which cannot be to be.

The standard view also imagines something which
cannot be to be. However, its proponents do not imagine
it to be in such a way that it forces contradiction.

WM

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Oct 12, 2013, 11:58:00 AM10/12/13

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On Saturday, 12 October 2013 16:34:12 UTC+2, fom wrote:
> On 10/12/2013 8:33 AM, WM wrote: > On Sunday, 6 October 2013 02:32:27 UTC+2,

> > If you want to well-order a set you have to index every element, such that every subset has a first element. How do you index uncountably many elements with only countably many numbers/indices?

> Your contradiction arises by imagining something which cannot be to be. The standard view also imagines something which cannot be to be. However, its proponents do not imagine it to be in such a way that it forces contradiction.

Its proponents did know that |R can be well-ordered. Zermelo wrote in 1904: Proof that every set can be well-ordered. Fraenkel in 1923 emphasized that it had *not yet* been possible to find a well-ordering.
Only after it had been proven that well-ordering of |R is impossible, the addicts of the infinite dropped that belief. They cannot accept a contradiction in their pet theory, because most of them had to recognize that they devoted their whole life to nonsense. Nevertheless, they did. Thousands of mathematicians did nothing but nonsense during their whole life.

Had anybody ever before made axioms in order to show that they are false? Look at these:

For every two points A and B there exists a line a that contains them both.

For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.

There exist at least two points on a line. There exist at least three points that do not lie on a line.

Should the following axiom be of completely different character?

For every two real numbers there exists one and only one relation <, =, >.

Only those poor matheologians who have wasted their lifetimes with Cantor-nonsense cannot accept the truth. That's a psychological problem, not a problem of mathematics.

Regards, WM

fom

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Oct 12, 2013, 12:34:08 PM10/12/13

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On 10/12/2013 10:58 AM, WM wrote:
> On Saturday, 12 October 2013 16:34:12 UTC+2, fom wrote:
>> On 10/12/2013 8:33 AM, WM wrote: > On Sunday, 6 October 2013 02:32:27 UTC+2,
>
>>> If you want to well-order a set you have to index every element, such that every subset has a first element. How do you index uncountably many elements with only countably many numbers/indices?
>
>> Your contradiction arises by imagining something which cannot be to be. The standard view also imagines something which cannot be to be. However, its proponents do not imagine it to be in such a way that it forces contradiction.
>
> Its proponents did know that |R can be well-ordered. Zermelo wrote in 1904: Proof that every set can be well-ordered. Fraenkel in 1923 emphasized that it had *not yet* been possible to find a well-ordering.
> Only after it had been proven that well-ordering of |R is impossible, the addicts of the infinite dropped that belief. They cannot accept a contradiction in their pet theory, because most of them had to recognize that they devoted their whole life to nonsense. Nevertheless, they did. Thousands of mathematicians did nothing but nonsense during their whole life.
>

Accepting a proof of such impossibility requires
a comparable act of imagined knowledge concerning
the continuum.

The issue here is known as independence.

In models of set theory a global well-ordering
implies that all sets are well-ordered. I do
not know enough about models of set theory without
choice, but the possibility certainly exists that
the reals do not have a well-ordering in models
without the explicit assumption of choice.

> Had anybody ever before made axioms in order to show that they are false? Look at these:
>
> For every two points A and B there exists a line a that contains them both.
>

This expresses sufficient knowledge to postulate a
line. It is not, however, the complete determination
of a line.

> For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.
>

This expresses that the sufficient knowledge to
postulate a line also suffices to segregate an
individual line. It is not, however, the complete
determination of a line.

> There exist at least two points on a line. There exist at least three points that do not lie on a line.
>

The first expresses that sufficient knowledge to
postulate a line is universally sufficient. Everything
that can be a line has the property needed to
postulate it.

The second expresses that the universe of lines
is plural.

Neither involves the complete determination of a

line.

> Should the following axiom be of completely different character?
>
> For every two real numbers there exists one and only one relation <, =, >.
>

This statement refers to the complete determination
of real numbers by virtue of the finite quantifier.

> Only those poor matheologians who have wasted their lifetimes with Cantor-nonsense cannot accept the truth. That's a psychological problem, not a problem of mathematics.
>

Well, if it is a psychological problem, the notion of
independence forces it to be symmetrical. It is your
psychological problem as well.

As for what is and what is not mathematics, that subject
has nothing to do with the reply I gave to your previous
post.

You asked a question about imagination -- both yours and
your opponents. That is what I answered.

If ever you wish to introduce me to One or Three, I
will be happy to meet with them. I think that possibility
just as unlikely as Virgil introducing me to Omega.

http://en.wikipedia.org/wiki/Empty_name

Virgil

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Oct 12, 2013, 4:40:34 PM10/12/13

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In article <70133969-fbfb-4ed3...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Sunday, 6 October 2013 02:32:27 UTC+2, Virgil wrote:
> > It is the analysis of the definable real number system, which necessarily
> > has more numbers in it than WM can define.
>
> If you want to well-order a set you have to index every element,

I have no wish to well-order the reals.

And I do not care whether or not such a well-ordring should even be
considered possible.
--

Virgil

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Oct 12, 2013, 4:44:34 PM10/12/13

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In article <8817f8bd-f870-4fd9...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Saturday, 12 October 2013 16:34:12 UTC+2, fom wrote:
> > On 10/12/2013 8:33 AM, WM wrote: > On Sunday, 6 October 2013 02:32:27
> > UTC+2,
>
> > > If you want to well-order a set you have to index every element, such
> > > that every subset has a first element. How do you index uncountably many
> > > elements with only countably many numbers/indices?
>
> > Your contradiction arises by imagining something which cannot be to be.
> > The standard view also imagines something which cannot be to be. However,
> > its proponents do not imagine it to be in such a way that it forces
> > contradiction.
>
> Its proponents did know that |R can be well-ordered. Zermelo wrote in 1904:
> Proof that every set can be well-ordered. Fraenkel in 1923 emphasized that it
> had *not yet* been possible to find a well-ordering.

A well-orderng of the reals depends critically on having assumed a
strong enough form of the axiom of choice.
I haven't.
--

WM

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Oct 13, 2013, 7:32:56 AM10/13/13

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On Saturday, 12 October 2013 18:34:08 UTC+2, fom wrote:

> the possibility certainly exists that the reals do not have a well-ordering in models without the explicit assumption of choice.

The certainty exists that an uncountable set (not the reals of mathematics) cannot have a well-ordering. In particular because there is no uncountable set. But even if it existed, there were not enough indices.

>> Had anybody ever before made axioms in order to show that they are false? Look at these:
>
>> For every two points A and B there exists a line a that contains them both.
>
> This expresses sufficient knowledge to postulate a line. It is not, however, the complete determination of a line.

But every two points can be known and determine a line completely.

> > For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.

> This expresses that the sufficient knowledge to postulate a line also suffices to segregate an individual line. It is not, however, the complete
determination of a line.

Given two points, the complete determination of the line is given.

Contrary to all these axioms: given |R, it cannot be well-ordered.

> You asked a question about imagination -- both yours and your opponents. That is what I answered.

No. I stated that the axiom of choice is the only axiom that explicitly claims something that is impossible. To accept that requires a psychological deformation.

Regards, WM

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Oct 13, 2013, 6:40:18 PM10/13/13

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In article <f29db4c5-768d-4a44...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Saturday, 12 October 2013 18:34:08 UTC+2, fom wrote:
>
> > the possibility certainly exists that the reals do not have a well-ordering
> > in models without the explicit assumption of choice.
>
> The certainty exists that an uncountable set (not the reals of mathematics)
> cannot have a well-ordering. In particular because there is no uncountable
> set. But even if it existed, there were not enough indices.

I am unaware of any property of well-ordeings that requires everything
to be indexable.

>
> Contrary to all these axioms: given |R, it cannot be well-ordered.

WM again makes claims without proofs.

> No. I stated that the axiom of choice is the only axiom that explicitly
> claims something that is impossible.

What does it claim that is impossible outside of WM's wild weird world
of WMytheology? ?
--

WM

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Oct 14, 2013, 12:44:25 PM10/14/13

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On Monday, 14 October 2013 00:40:18 UTC+2, Virgil wrote:

> I am unaware of any property of well-ordeings that requires everything to be indexable.

What do you well-order, if not identifyable elements? How do you identify elements?

>> I stated that the axiom of choice is the only axiom that explicitly claims something that is impossible.

> What does it claim that is impossible

Ordering of elements implies identification of these elements. That is impossible without knowing the elements.

> >> A well-orderng of the reals depends critically on having assumed a strong enough form of the axiom of choice.
>
>> No.

> Are you claiming that without any axiom of choice one can prove that
reals to be well-orderable?

No, I know that it is impossible to well-order elements that cannot be identified. No axiom does help in that case. And none of the alleged "proofs" of matheology can change anything.

Regards, WM

Virgil

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Oct 14, 2013, 4:19:20 PM10/14/13

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In article <d0a7e7c5-9278-4cb1...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Monday, 14 October 2013 00:40:18 UTC+2, Virgil wrote:
>
> > I am unaware of any property of well-ordeings that requires everything to
> > be indexable.
>
> What do you well-order, if not identifyable elements?

All elements, whether "identifyable" in WM's sense, or not.
For a set to be well-ordered, the ordering must apply to all its
elements, not just the few that WM acknowledges.

> How do you identify
> elements?

They do it themselves, if they can.

>
> >> I stated that the axiom of choice is the only axiom that explicitly claims
> >> something that is impossible.
>
> > What does it claim that is impossible
>
> Ordering of elements implies identification of these elements. That is
> impossible without knowing the elements.

An well-ordering relation can be defined on some sets, like the
naturals, without identifying individually more than one member.
--

WM

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Oct 15, 2013, 6:46:42 AM10/15/13

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On Monday, 14 October 2013 22:19:20 UTC+2, Virgil wrote:
> > How do you identify > elements?

> They do it themselves, if they can.

Can they? Can they inform you about their position? And if they cannot?

> An well-ordering relation can be defined on some sets, like the naturals, without identifying individually more than one member.

Only if every element can be identified. This is not possible for undefinable real numbers. Unless they know their place. They are the angels of matheology.

Regards, WM

Virgil

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Oct 15, 2013, 6:58:32 PM10/15/13

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In article <8cb2fd38-0b55-4ba9...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Monday, 14 October 2013 22:19:20 UTC+2, Virgil wrote:
> > > How do you identify > elements?
>
> > They do it themselves, if they can.
>
> Can they? Can they inform you about their position? And if they cannot?

They certainly do not seem to be able to tell WM anything,
but reality does not seem able to tell WM anything either.

>
> > An well-ordering relation can be defined on some sets, like the naturals,
> > without identifying individually more than one member.
>
> Only if every element can be identified.

Then the natural numbers can not be, in WM's view, well-ordered,
because WM says HE cannot identify each and every one of them.

He could actually do this if, but only if, WMe admitted an actually
infinite set of naturals.
--

WM

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Oct 17, 2013, 9:27:24 AM10/17/13

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On Wednesday, 16 October 2013 00:58:32 UTC+2, Virgil wrote:
> In article <8cb2fd38-0b55-4ba9...@googlegroups.com>,
>
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>
>
> > On Monday, 14 October 2013 22:19:20 UTC+2, Virgil wrote:
>
> > > > How do you identify > elements?
>
> >
>
> > > They do it themselves, if they can.
>
> >
>
> > Can they? Can they inform you about their position? And if they cannot?
>
>
>
> They certainly do not seem to be able to tell WM anything,
>

O, I knew it. First you must become enlightened. That is usually required in theology.

>
> >
>
> > > An well-ordering relation can be defined on some sets, like the naturals,
>
> > > without identifying individually more than one member.
>
> >
>
> > Only if every element can be identified.
>
>
>
> Then the natural numbers can not be, in WM's view, well-ordered,
>
> because WM says HE cannot identify each and every one of them.

On the contrary. A number that I could not identify, in principle, in decimal representation is not a natural numbers.

>
>
>
> He could actually do this if, but only if, WMe admitted an actually
>
> infinite set of naturals.

I do not prohibit them. They simply never appear anywhere.

Regards, WM

Wisely Non-Theist

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Oct 17, 2013, 7:11:31 PM10/17/13

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> On Wednesday, 16 October 2013 00:58:32 UTC+2, Virgil wrote:
> > In article <8cb2fd38-0b55-4ba9...@googlegroups.com>,
> > WM <muec...@rz.fh-augsburg.de> wrote:
> > > On Monday, 14 October 2013 22:19:20 UTC+2, Virgil wrote:
> > > > > How do you identify > elements?
> > > > They do it themselves, if they can.
> > > Can they? Can they inform you about their position? And if they cannot?
> > They certainly do not seem to be able to tell WM anything,
>
> O, I knew it. First you must become enlightened.

Enlightened far beyond WM's capacity for elightenment.

He seems to maor in endarkenment.

> That is usually required in
> theology.

Theology would be a better major for Wm as it, like he, is based only on
faith, not ever on fact.

> > > > An well-ordering relation can be defined on some sets, like the
> > > > naturals,
> > > > without identifying individually more than one member.
> > > Only if every element can be identified.
> > Then the natural numbers can not be, in WM's view, well-ordered,
> > because WM says HE cannot identify each and every one of them.
>
> On the contrary. A number that I could not identify, in principle, in decimal
> representation is not a natural numbers.

Does WM claim to be able to identify each and everyone of them , that is
to say ALL of them?

> > He could actually do this if, but only if, WMe admitted an actually
> > infinite set of naturals.
>
> I do not prohibit them. They simply never appear anywhere.

But they do appear collectively, and as such they form a set.
At least they do everywhere outside of WM's wild weird wet world of
WMytheology.

>
> Regards, WM

WM

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Oct 18, 2013, 8:29:08 AM10/18/13

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On Friday, 18 October 2013 01:11:31 UTC+2, Wisely Non-Theist wrote:

> > > > > > How do you identify elements?
> > > > > They do it themselves, if they can.
> > > > Can they? Can they inform you about their position? And if they cannot?
> > > They certainly do not seem to be able to tell WM anything,
> > O, I knew it. First you must become enlightened.
> Enlightened far beyond WM's capacity for elightenment.

And far beyond that of others like Wittgenstein, Weyl, Brouwer, Poincaré, Zeilberger and so on.

We fix this fact: There is no way to show how the elements do something themselves.

> That is usually required in
> theology.

> > On the contrary. A number that I could not identify, in principle, in decimal
> > representation is not a natural numbers.
> Does WM claim to be able to identify each and everyone of them , that is to say ALL of them?

No, "every one" does not lead to a completed "all" in infinite sets.

> > I do not prohibit them. They simply never appear anywhere.
> But they do appear collectively, and as such they form a set.

They do not appear. The idea of a set of them appears - in matheological delusions.

Regards, WM

fom

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Oct 18, 2013, 11:04:48 AM10/18/13

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On 10/18/2013 7:29 AM, WM wrote:
> On Friday, 18 October 2013 01:11:31 UTC+2, Wisely Non-Theist wrote:
>
>
>>>>>>> How do you identify elements?
>>>>>> They do it themselves, if they can.
>>>>> Can they? Can they inform you about their position? And if they cannot?
>>>> They certainly do not seem to be able to tell WM anything,
>>> O, I knew it. First you must become enlightened.
>> Enlightened far beyond WM's capacity for elightenment.
>
> And far beyond that of others like Wittgenstein, Weyl, Brouwer, Poincaré, Zeilberger and so on.
>
> We fix this fact: There is no way to show how the elements do something themselves.
>

What you actually show is this:

Suppose we consider the collection of statements,

{Ex(~(x=1)), Ex(~(x=2)), Ex(~(x=3)), ...}

Your claim is that every finite conjunction of
these statements true.

Your claim is that the possibility of each
such statement being true simultaneously is
false.

This is in violation with the compactness theorem
from first-order logic.

As it is derived from the completeness theorem, you
are claiming that there is no complete logic.

But, without a complete logic, counterexamples do
not necessarily invalidate assertions.

This is not necessarily wrong. The Brouwerian
idea is that construction proceeds indefinitely
until contradiction arises within the construction.

I suppose then, that there is no sense of an
independent mathematics external to any given
construction.

Wisely Non-Theist

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Oct 18, 2013, 3:48:41 PM10/18/13

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In article <09d908cd-f12a-42d5...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Friday, 18 October 2013 01:11:31 UTC+2, Wisely Non-Theist wrote:
>
>
> > > > > > > How do you identify elements?
> > > > > > They do it themselves, if they can.
> > > > > Can they? Can they inform you about their position? And if they
> > > > > cannot?
> > > > They certainly do not seem to be able to tell WM anything,
> > > O, I knew it. First you must become enlightened.
> > Enlightened far beyond WM's capacity for elightenment.
>
> And far beyond that of others like Wittgenstein, Weyl, Brouwer, Poincaré,
> Zeilberger and so on.
>
> We fix this fact: There is no way to show how the elements do something
> themselves.
>
> > That is usually required in
> > theology.
> > > On the contrary. A number that I could not identify, in principle, in
> > > decimal
> > > representation is not a natural numbers.
> > Does WM claim to be able to identify each and everyone of them , that is to
> > say ALL of them?
>
> No, "every one" does not lead to a completed "all" in infinite sets.

If for some sets WM claims that every one "does not lead to" all,
which members of such sets does WM claim it does NOT lead to?

>
> > > I do not prohibit them. They simply never appear anywhere.
> > But they do appear collectively, and as such they form a set.
>
> They do not appear.

You mean a set can be an infinite set without having infinitely many
members?

> The idea of a set of them appears - in matheological

> delusions. Then, since ther is no such thing as a set of inatuals or a set of integers or a
set of rationals or a set of reals, ther is no way to distinguish
between natuals integers rationals and reals, nor any way to to
determine whetehr any of those definitions are instantiated.

At least no way in WM's wild weird world of WMytheology/