Alan Mackenzie schrieb am Samstag, 26. März 2022 um 11:43:58 UTC+1:
> WM <askas...@gmail.com> wrote:

> > Every written sequence necessarily ends somewhere. But those who
> > believe that infinite sequences could exist without generating formula
> > should know how to continue. What is the next digit following upon
> > 0.322322322322322322322 in an infinite sequence?
> Which infinite sequence were you thinking of? We can't read your mind
> (thankfully).

If you insist that an infinite sequence without generating formula defines a real number you must read whatever, because you cannot know more than finitely many digits. See above: Every written sequence necessarily ends somewhere.
>
> If you are thinking about that sequence being the decimal expansion of a
> real number N, then the next digit is the integer part of ((N -
> 0.322322322322322322322) * 10^22).

But you cannot know which real numbers N without knowing it, i.e., without knowing the generating formula.

Regards, WM

FromTheRafters schrieb am Samstag, 26. März 2022 um 11:44:30 UTC+1:
> WM explained on 3/26/2022 :

> > It is easy to tell what you are thinking. You claim that an infinite sequence
> > can be given without formula.
> Not at all, I'm saying such a formula describes a particular number,
> but its CDE representation does not define the reals. There is a home
> for every real,

If you wish to apply Cantor's diagonal argument, you must believe that infinite digit sequences define individual real numbers. On the other hand only the first part, from the first digit to the diagonal digit, is relevant. Therefore the argument is void with respect to irrationals. Cantor's list contains only rational numbers.

Regards, WM

Alan Mackenzie schrieb am Samstag, 26. März 2022 um 11:57:25 UTC+1:

> Perhaps you might like to expand on what you think you mean by the
> non-existence of such infinite decimal representations.

It is hard to believe that you are unsure about it. Simplest example: If you write 1/3, then we know that 0.333... will never contain any other digit that 3. If you write
0.3333333333333333333333333333333, giving as many digits as you like, we cannot be sure that you mean 1/3.

Regards, WM

Eram semper recta schrieb am Samstag, 26. März 2022 um 13:08:57 UTC+1:
> On Saturday, 26 March 2022 at 11:48:39 UTC+2, WM wrote:

> > I did not say so. I said that there is no infinite decimal representation without a formula prescribing every digit. 1/4 and 1/3 are such formulas..
> Neither 1/4 nor 1/3 are formulas.

I call them formulas because they define the digits of their decimal representations.

> They are complete well-formed numbers. They do not have to be represented in any particular radix system.

In principle they are instructions what to do: divide 1 by 3, if impossible multiply 1 by 10, write the result.

Regards, WM

david...@gmail.com schrieb am Sonntag, 27. März 2022 um 18:48:34 UTC+2:
> On Saturday, March 19, 2022 at 9:35:13 AM UTC-7, WM wrote:
> > There is no permutation of the Xs in
> >
> > XOOO...
> > XOOO...
> > XOOO...
> > XOOO...
> > ...
> >
> > such that all positions are covered by Xs.
>
> Would you agree that there is permutation of the X's and 0's such that all the 0's end up in the first column and all the other columns are "covered" by X's?

No, that is obviously wrong. Here are the fractions m/n written out:

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
....

The sequence described by Cantor's formula
k = (m + n - 1)(m + n - 2)/2 + m
is
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...

When it is gathered in the first column, exchanging the fractions with the fractions residing there, then we get

1/1, 2/1, 1/3, 1/4, ... 1/1, 3/1, 1/3, 1/4, ... 1/1, 3/1, 4/1, 1/4, ... 1/1, 3/1, 4/1, 1/4, ...
1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 5/1, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ... 4/1, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 2/2, 5/2, 5/3, 5/4, ...
.... ... ... ...

Never, in no finite step, the number of fractions residing outside of the first column will be reduced. And after all finite steps nothing happens in mathematics. Therefore not all fractions are enumerated = gathered in the first column.

If Cantor were right, then the result would look like this

1/1, __, __, __, ...
1/2, __, __, __, ...
2/1, __, __, __, ...
1/3, __, __, __, ...
2/2, __, __, __, ...
....

But never a place becomes empty.
>
>
>
> > If it were, it would imply, that in the sequence
> >
> > XOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
> >
> > all Os could be covered by the X.
>
> I don't see the logic to that.

Every line of the matrix has this shape. Therefore it is impossible by shifting X in other lines to cover the amtrix.
>
>
> > Therefore Cantor's notion of countability is trash.
>
> I don't disagree with your conclusion, but I simply don't understand the argument you use to arrive at that conclusion. Do you think you have a better argument than the one Poincare gave us?
>
>
> Let me remind everybody about the argument that Poincare (and others) made:
>
> 1) Above all else, mathematics is a language. That is, it's a system of symbols and rules for manipulating strings of symbols.
>
> 2) The purpose of the language is to give us a way to encode and communicate "ideas".
>
> 3) The things that can be said to "exist" in mathematics are ideas! Or in other words, the things that "exist" in the language of mathematics are things we can actually talk about, and identify, and describe.
>
> 4) Emphatically, ideas can be completely defined in a finite number of words of the language.
>
> 5) If we take Cantor's ideas about infinite sets seriously, we would conclude that there cannot be more than "countably" many ideas, since there are not more than "countably" many finite strings of symbols.
>
> 6) But then Cantor goes on to "prove" that there "exist" sets (collections of objects) that are bigger than any "countable" collection. And he claims that this "proves" that there "exist" objects in mathematics that are not "ideas" (and hence don't actually "exist")
>
> 7) That last conclusion is so absurd that the whole idea of "countable" should be thrown away. It cannot be relevant to the important mathematics that has applications in reasoning about the real world.

My argument is not better, but it shows in a soimple way that the fractions cannot be counted. Therefore uncountability is nonsense too. Of course (7) is clearly true. Never any argument abot countable sets has made sense in mathematics.

Regards, WM

Jim Burns schrieb am Sonntag, 27. März 2022 um 13:27:56 UTC+2:

> I don't know what the next digit is.
> _If you're telling the truth_ about it being
> an infinite digit sequence,
> then it _has_ a next digit.

Yes, we know how the set of real numbers is defined. But you cannot define all individuals. For that sake you need a formula.

Regards, WM

On 3/27/2022 5:35 PM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 27. März 2022 um 13:27:56 UTC+2:

>> I don't know what the next digit is.
>> _If you're telling the truth_ about it being
>> an infinite digit sequence,
>> then it _has_ a next digit.
>
> Yes, we know how the set of real numbers is defined.
> But you cannot define all individuals.
> For that sake you need a formula.

Consider the possibility that we don't particularly
want to define all individuals.

We want to make claims about a real number
which we know are true
without needing to choose which real number
is referred to.

It doesn't look like we need to define all
real numbers in order to do that.

One very nice feature of these claims-known-to-be-
-true-without-needing-to-choose is that
we can not-choose among _infinitely-many_ and thus
make true claims about each of _infinitely-many_
even though we ourselves are _finite_

On Monday, 28 March 2022 at 00:18:54 UTC+3, WM wrote:
> Eram semper recta schrieb am Samstag, 26. März 2022 um 13:08:57 UTC+1:
> > On Saturday, 26 March 2022 at 11:48:39 UTC+2, WM wrote:
>
> > > I did not say so. I said that there is no infinite decimal representation without a formula prescribing every digit. 1/4 and 1/3 are such formulas.
> > Neither 1/4 nor 1/3 are formulas.
> I call them formulas because they define the digits of their decimal representations.

Whilst they certainly influence the decimal representations, they do NOT define the digits of their decimal representations. Measure defines the representations.

> > They are complete well-formed numbers. They do not have to be represented in any particular radix system.

> In principle they are instructions what to do: divide 1 by 3, if impossible multiply 1 by 10, write the result.

Not at all. They are *NAMES* given to completed measures. There is nothing pending.

A number is a name given to a measure that describes a magnitude or size. In particular, the magnitude being described is the antecedent part of a ratio of magnitudes.

_ : _ _
_ _ : _ _ _ _
_ _ _ : _ _ _ _ _ _

All the above are ratios of line segments (spaces only included for clarity) whose MEASURE is 1/2. The above ratios are in proportion, but they re not equal. Their measure is however equal. Once measured, the measure is given a name, that is, one half or 1/2 or 2/4, etc, but there is no process or formula pending anywhere. The measure is in the past. It is history.

Now the conversion to decimal is another (secondary) process whereby the properties of the number obviously influence the decimal measure, but calling the name the same thing as the process is absurd.

Analogy:
Cream is used to make ice cream, but we do not call cream an "ice-cream maker".
p/q is used to find a decimal measure, but we do not call p/q the decimal measure.

p/q (number) is not the conversion (decimal measure). Do you understand now?

I have always stated that unless you first become a philosopher, you cannot be a mathematician.

Till you gain this wisdom which I impart to you, you'll keep saying nonsense like you said to Jim Burns:

"Yes, we know how the set of real numbers is defined. But you cannot define all individuals. For that sake you need a formula."

If Jim Burns even had a modicum of brains, then he would have responded:

"But you just said p/q ( "real number") is the process definition (formula)", which shows that you contradict yourself all the time.

Again: Number =/= Formula. Look up the meaning of these words in the dictionary - it might help you!

>
> Regards, WM

On Monday, 28 March 2022 at 07:48:59 UTC+3, Eram semper recta wrote:
> On Monday, 28 March 2022 at 00:18:54 UTC+3, WM wrote:
> > Eram semper recta schrieb am Samstag, 26. März 2022 um 13:08:57 UTC+1:
> > > On Saturday, 26 March 2022 at 11:48:39 UTC+2, WM wrote:
> >
> > > > I did not say so. I said that there is no infinite decimal representation without a formula prescribing every digit. 1/4 and 1/3 are such formulas.
> > > Neither 1/4 nor 1/3 are formulas.
> > I call them formulas because they define the digits of their decimal representations.
> Whilst they certainly influence the decimal representations, they do NOT define the digits of their decimal representations. Measure defines the representations.

Example: 1/4 in base 10: ***1/4 x 10/10 = 10/4 x 1/10 = (2 + 1/2) x 1/10 = (2 x 1/10)+ (10/2 x 1/100)= (2 x 1/10)+(5 x 1/100) = 0.25 ***

1/4 in base 12: *** 1/4 x 12/12 = 12/4 x 1/12 = 3 x 112 = 0.3 ***

Can you see how different are the starred processes (Formulas which produce each DIGIT!)?

The number is NOTTTTTTT the formula. Lerne das gut!

> > > They are complete well-formed numbers. They do not have to be represented in any particular radix system.
>
> > In principle they are instructions what to do: divide 1 by 3, if impossible multiply 1 by 10, write the result.
> Not at all. They are *NAMES* given to completed measures. There is nothing pending.
>
> A number is a name given to a measure that describes a magnitude or size. In particular, the magnitude being described is the antecedent part of a ratio of magnitudes.
>
> _ : _ _
> _ _ : _ _ _ _
> _ _ _ : _ _ _ _ _ _
>
> All the above are ratios of line segments (spaces only included for clarity) whose MEASURE is 1/2. The above ratios are in proportion, but they re not equal. Their measure is however equal. Once measured, the measure is given a name, that is, one half or 1/2 or 2/4, etc, but there is no process or formula pending anywhere. The measure is in the past. It is history.
>
> Now the conversion to decimal is another (secondary) process whereby the properties of the number obviously influence the decimal measure, but calling the name the same thing as the process is absurd.
>
> Analogy:
> Cream is used to make ice cream, but we do not call cream an "ice-cream maker".
> p/q is used to find a decimal measure, but we do not call p/q the decimal measure.
>
> p/q (number) is not the conversion (decimal measure). Do you understand now?
>
> I have always stated that unless you first become a philosopher, you cannot be a mathematician.
>
> Till you gain this wisdom which I impart to you, you'll keep saying nonsense like you said to Jim Burns:
> "Yes, we know how the set of real numbers is defined. But you cannot define all individuals. For that sake you need a formula."
> If Jim Burns even had a modicum of brains, then he would have responded:
>
> "But you just said p/q ( "real number") is the process definition (formula)", which shows that you contradict yourself all the time.
>
> Again: Number =/= Formula. Look up the meaning of these words in the dictionary - it might help you!
>
> >
> > Regards, WM

On Monday, 28 March 2022 at 07:56:45 UTC+3, Eram semper recta wrote:
> On Monday, 28 March 2022 at 07:48:59 UTC+3, Eram semper recta wrote:
> > On Monday, 28 March 2022 at 00:18:54 UTC+3, WM wrote:
> > > Eram semper recta schrieb am Samstag, 26. März 2022 um 13:08:57 UTC+1:
> > > > On Saturday, 26 March 2022 at 11:48:39 UTC+2, WM wrote:
> > >
> > > > > I did not say so. I said that there is no infinite decimal representation without a formula prescribing every digit. 1/4 and 1/3 are such formulas.
> > > > Neither 1/4 nor 1/3 are formulas.
> > > I call them formulas because they define the digits of their decimal representations.
> > Whilst they certainly influence the decimal representations, they do NOT define the digits of their decimal representations. Measure defines the representations.
> Example: 1/4 in base 10: ***1/4 x 10/10 = 10/4 x 1/10 = (2 + 1/2) x 1/10 = (2 x 1/10)+ (10/2 x 1/100)= (2 x 1/10)+(5 x 1/100) = 0.25 ***
>
> 1/4 in base 12: *** 1/4 x 12/12 = 12/4 x 1/12 = 3 x 112 = 0.3 ***
>
> Can you see how different are the starred processes (Formulas which produce each DIGIT!)?

The processes (measures) are DIFFERENT because in one case we use tenths or exponents of tenths and in the other case we use twelfths or exponents of 12. The number is only an INPUT to the process which clearly tells us that the number and the process (formula) are two different concepts.

>
> The number is NOTTTTTTT the formula. Lerne das gut!
> > > > They are complete well-formed numbers. They do not have to be represented in any particular radix system.
> >
> > > In principle they are instructions what to do: divide 1 by 3, if impossible multiply 1 by 10, write the result.
> > Not at all. They are *NAMES* given to completed measures. There is nothing pending.
> >
> > A number is a name given to a measure that describes a magnitude or size. In particular, the magnitude being described is the antecedent part of a ratio of magnitudes.
> >
> > _ : _ _
> > _ _ : _ _ _ _
> > _ _ _ : _ _ _ _ _ _
> >
> > All the above are ratios of line segments (spaces only included for clarity) whose MEASURE is 1/2. The above ratios are in proportion, but they re not equal. Their measure is however equal. Once measured, the measure is given a name, that is, one half or 1/2 or 2/4, etc, but there is no process or formula pending anywhere. The measure is in the past. It is history.
> >
> > Now the conversion to decimal is another (secondary) process whereby the properties of the number obviously influence the decimal measure, but calling the name the same thing as the process is absurd.
> >
> > Analogy:
> > Cream is used to make ice cream, but we do not call cream an "ice-cream maker".
> > p/q is used to find a decimal measure, but we do not call p/q the decimal measure.
> >
> > p/q (number) is not the conversion (decimal measure). Do you understand now?
> >
> > I have always stated that unless you first become a philosopher, you cannot be a mathematician.
> >
> > Till you gain this wisdom which I impart to you, you'll keep saying nonsense like you said to Jim Burns:
> > "Yes, we know how the set of real numbers is defined. But you cannot define all individuals. For that sake you need a formula."
> > If Jim Burns even had a modicum of brains, then he would have responded:
> >
> > "But you just said p/q ( "real number") is the process definition (formula)", which shows that you contradict yourself all the time.
> >
> > Again: Number =/= Formula. Look up the meaning of these words in the dictionary - it might help you!
> >
> > >
> > > Regards, WM

On 3/27/2022 4:12 PM, WM wrote:
> FromTheRafters schrieb am Samstag, 26. März 2022 um 11:44:30 UTC+1:
>> WM explained on 3/26/2022 :
>
>>> It is easy to tell what you are thinking. You claim that an infinite sequence
>>> can be given without formula.
>> Not at all, I'm saying such a formula describes a particular number,
>> but its CDE representation does not define the reals. There is a home
>> for every real,
>
> If you wish to apply Cantor's diagonal argument, you must believe that infinite digit sequences define individual real numbers. On the other hand only the first part, from the first digit to the diagonal digit, is relevant. Therefore the argument is void with respect to irrationals. Cantor's list contains only rational numbers.
>
> Regards, WM

Your own words show you do not understand Cantor at all.

On 3/27/2022 4:08 PM, WM wrote:
> Alan Mackenzie schrieb am Samstag, 26. März 2022 um 11:43:58 UTC+1:
>> WM <askas...@gmail.com> wrote:
>
>>> Every written sequence necessarily ends somewhere. But those who
>>> believe that infinite sequences could exist without generating formula
>>> should know how to continue. What is the next digit following upon
>>> 0.322322322322322322322 in an infinite sequence?
>> Which infinite sequence were you thinking of? We can't read your mind
>> (thankfully).
>
> If you insist that an infinite sequence without generating formula defines a real number you must read whatever, because you cannot know more than finitely many digits.

Wrong thinking, you are describing "External User Dependent Math", where the Math depends upon who is looking at it.
This is both silly and stupid.

See above: Every written sequence necessarily ends somewhere.
>>
>> If you are thinking about that sequence being the decimal expansion of a
>> real number N, then the next digit is the integer part of ((N -
>> 0.322322322322322322322) * 10^22).
>
> But you cannot know which real numbers N without knowing it, i.e., without knowing the generating formula.

no problem, what are you going to use the number for ?

>
> Regards, WM

On 3/27/2022 4:15 PM, WM wrote:
> Alan Mackenzie schrieb am Samstag, 26. März 2022 um 11:57:25 UTC+1:
>
>> Perhaps you might like to expand on what you think you mean by the
>> non-existence of such infinite decimal representations.
>
> It is hard to believe that you are unsure about it. Simplest example: If you write 1/3, then we know that 0.333... will never contain any other digit that 3. If you write
> 0.3333333333333333333333333333333, giving as many digits as you like, we cannot be sure that you mean 1/3.
>
> Regards, WM

try to be clear;

0.333... = 1/3

0.333 is not equal to 1/3

On 3/27/2022 4:18 PM, WM wrote:
> Eram semper recta schrieb am Samstag, 26. März 2022 um 13:08:57 UTC+1:
>> On Saturday, 26 March 2022 at 11:48:39 UTC+2, WM wrote:
>
>>> I did not say so. I said that there is no infinite decimal representation without a formula prescribing every digit. 1/4 and 1/3 are such formulas.
>> Neither 1/4 nor 1/3 are formulas.
>
>
> I call them formulas because they define the digits of their decimal representations.

Wrong. They are fractions or rationals. go back to school.

>
>> They are complete well-formed numbers. They do not have to be represented in any particular radix system.
>
> In principle they are instructions what to do: divide 1 by 3, if impossible multiply 1 by 10, write the result.

You never took Algebra.

>
> Regards, WM

On Monday, 28 March 2022 at 00:12:25 UTC+3, WM wrote:
> FromTheRafters schrieb am Samstag, 26. März 2022 um 11:44:30 UTC+1:
> > WM explained on 3/26/2022 :
> > > It is easy to tell what you are thinking. You claim that an infinite sequence
> > > can be given without formula.
> > Not at all, I'm saying such a formula describes a particular number,
> > but its CDE representation does not define the reals. There is a home
> > for every real,
> If you wish to apply Cantor's diagonal argument, you must believe that infinite digit sequences define individual real numbers.

I said this many decades ago and was laughed to scorn. Cantor was right that the set of real numbers are uncountable because "real numbers" don't exist (as any well-formed concept) - meaning they can't all be defined and you can't count those objects you don't know about. You finally realised this - bravo!

> On the other hand only the first part, from the first digit to the diagonal digit, is relevant. Therefore the argument is void with respect to irrationals. Cantor's list contains only rational numbers.
>
> Regards, WM

Eram semper recta schrieb am Montag, 28. März 2022 um 06:48:59 UTC+2:
> On Monday, 28 March 2022 at 00:18:54 UTC+3, WM wrote:

> > > They are complete well-formed numbers. They do not have to be represented in any particular radix system.
>
> > In principle they are instructions what to do: divide 1 by 3, if impossible multiply 1 by 10, write the result.
> Not at all.

1/3 in words means "one divided by three".

> They are *NAMES* given to completed measures.

After the original meaning has been forgotten, they appear as names. Look, how many Millers are living in the USA? How many of them are producing flour?

> "Yes, we know how the set of real numbers is defined. But you cannot define all individuals. For that sake you need a formula."
> If Jim Burns even had a modicum of brains, then he would have responded:
>
> "But you just said p/q ( "real number") is the process definition (formula)", which shows that you contradict yourself all the time.

p/q is not a formula defining an individual fraction, like 1/3, but defining the general fraction.

Regards, WM

Jim Burns schrieb am Montag, 28. März 2022 um 00:40:14 UTC+2:
> On 3/27/2022 5:35 PM, WM wrote:

> > Yes, we know how the set of real numbers is defined.
> > But you cannot define all individuals.
> > For that sake you need a formula.
> Consider the possibility that we don't particularly
> want to define all individuals.

If you want to map every individual then you must define every indivdual.

Regards, WM

On 3/28/2022 11:57 AM, WM wrote:
> Eram semper recta schrieb am Montag, 28. März 2022 um 06:48:59 UTC+2:
>> On Monday, 28 March 2022 at 00:18:54 UTC+3, WM wrote:
>
>>>> They are complete well-formed numbers. They do not have to be represented in any particular radix system.
>>
>>> In principle they are instructions what to do: divide 1 by 3, if impossible multiply 1 by 10, write the result.
>> Not at all.
>
> 1/3 in words means "one divided by three".
>
>> They are *NAMES* given to completed measures.
>
> After the original meaning has been forgotten, they appear as names. Look, how many Millers are living in the USA? How many of them are producing flour?

only 1/3 of them

>
>> "Yes, we know how the set of real numbers is defined. But you cannot define all individuals. For that sake you need a formula."
>> If Jim Burns even had a modicum of brains, then he would have responded:
>>
>> "But you just said p/q ( "real number") is the process definition (formula)", which shows that you contradict yourself all the time.
>
> p/q is not a formula defining an individual fraction, like 1/3, but defining the general fraction.
>
> Regards, WM

p and q are "variables", which WM denies existence of variables. [it is for trolling]

On 3/28/2022 12:03 PM, WM wrote:
> Jim Burns schrieb am Montag, 28. März 2022 um 00:40:14 UTC+2:
>> On 3/27/2022 5:35 PM, WM wrote:
>
>>> Yes, we know how the set of real numbers is defined.
>>> But you cannot define all individuals.
>>> For that sake you need a formula.
>> Consider the possibility that we don't particularly
>> want to define all individuals.
>
> If you want to map every individual then you must define every indivdual.
>
> Regards, WM

does that apply to numbers ? or are you still Out Of Math (OOM)?

On 3/27/2022 12:48 PM, David Petry wrote:

> 2) The purpose of the language is to give us a way to
> encode and communicate "ideas".

Very often, language refers to things other than ideas.
There is the idea of a sheep, but there is also a sheep.
Language more often refers to a sheep, it seems to me,
but it can refer to either.

> 3) The things that can be said to "exist" in mathematics
> are ideas! Or in other words, the things that "exist" in
> the language of mathematics are things we can actually
> talk about, and identify, and describe.

I don't think you should have said "in other words".
Being an idea and being a topic of conversation
are not the same.

----
Language can make _indefinite references_ which can be
used to make claims about _one of_ various sheep or
of other things, without needing to choose which one.

We can make claims describing one of the things which
our description is true of, and we will know that our
claim is true -- without needing to choose which one.

This is so obvious that it might seem like I'm less than
serious here, but I think that this so-obvious fact
explains how, finite though we are, we can describe and
reason about infinitely-many things.

Describe a geometric line.
| | For each cut, there is a point marking the cut.
| | A cut LEFT,RIGHT partitions the points of the line
| so that each point in LEFT is to the left of
| each point in RIGHT
| | A point c marks LEFT,RIGHT iff
| each point to the left of c is in LEFT and
| each point to the right of c is in RIGHT

Our language describes infinitely-many cuts, infinitely-
-many points. But the language we use to do that
is finitely-long. Indefinite reference can do that.

> 4) Emphatically, ideas can be completely defined in
> a finite number of words of the language.

Emphatically, that makes a lovely eventual goal.
It sounds a lot like the comic instructions for
becoming vastly wealthy on Wall Street:
"Buy low and sell high".

Yesbut, what we want to hear is _how to do that_

Note that a finite number of words can describe infinitely-
-many things.

This is easy. The shorter the description, the more things
satisfy the the description.
However many red cars with a rosary hung from the rear-view
mirror exist, there are more red cars.
However many red cars exist, there are more cars.
Some descriptions are _short enough_ that they describe
infinitely-many things.

> 5) If we take Cantor's ideas about infinite sets seriously,
> we would conclude that there cannot be more than "countably"
> many ideas, since there are not more than "countably" many
> finite strings of symbols.

Not everything referred to is an idea.
Cuts of a geometric line and points in a geometric line
can also be referred to.

We have some finite-length claims about a _cut_ of
a geometric line. This is part of what we mean by
"geometric line". We could deny those claims, but then
we would mean something different by "geometric line".

Consider an infinite sequence of points in a geometric
line. For that sequence of points, there is a sequence
of nested intervals which partitions all the points in
the sequence into LEFT and RIGHT with none of its points
in between. Our description of a geometric line says
there is a point in the line in between, so the infinite
sequence does not contain all points in the line.

We use an indefinite reference to an infinite sequence
of points. We conclude that _any_ infinite sequence of
points does not contain all points in the line.

Suppose one objected to this conclusion.
One could change the description of a geometric line.
Instead of
| For each cut, there is a point marking the cut.

we could say
| For some cuts, no point marks the cut.

One consequence of this different description is that
two curves, both continuous _at every point_
might cross without intersecting.
But this isn't what we mean by geometric line.
We would have changed the description so that
we are no longer describing a geometric line.

These are the choices in front of us:
(i)
Say: "Yuck. I didn't want undefinable points.
But it is what it is."
(ii)
Say: "Clearly, geometric lines are not for me.
I will go and play solitaire.."

On 3/28/2022 1:03 PM, WM wrote:
> Jim Burns schrieb
> am Montag, 28. März 2022 um 00:40:14 UTC+2:
>> On 3/27/2022 5:35 PM, WM wrote:

>>> Yes, we know how the set of real numbers is defined.
>>> But you cannot define all individuals.
>>> For that sake you need a formula.
>>
>> Consider the possibility that we don't particularly
>> want to define all individuals.
>
> If you want to map every individual
> then you must define every indivdual.

We describe the objects we intend to study.

Here, we intend to study a geometric line.
We describe it as having the _Archimedean property_
and _Dedekind completeness_

If your dark realm does not satisfy the description,
we haven't forgotten it,
it's not part of what we intend to study.

One consequence of a line having the Archimedean property
and Dedekind completeness is that there is no less than one
and no more than one point which is in all intervals
represented by an infinite digit sequence.

That's
no less than one point, definable or not definable,
no more than one point, definable or not definable.

Your argument is that
Cantor is wrong because
his system implies undefinable points,
so, if his system is used,
there will be points undefinable.

Perhaps I missed it, but
you've never answered the question
"and then what?"

On Monday, 28 March 2022 at 19:57:06 UTC+3, WM wrote:
> Eram semper recta schrieb am Montag, 28. März 2022 um 06:48:59 UTC+2:
> > On Monday, 28 March 2022 at 00:18:54 UTC+3, WM wrote:
>
> > > > They are complete well-formed numbers. They do not have to be represented in any particular radix system.
> >
> > > In principle they are instructions what to do: divide 1 by 3, if impossible multiply 1 by 10, write the result.
> > Not at all.
> 1/3 in words means "one divided by three".
> > They are *NAMES* given to completed measures.
> After the original meaning has been forgotten, they appear as names. Look, how many Millers are living in the USA? How many of them are producing flour?
> > "Yes, we know how the set of real numbers is defined. But you cannot define all individuals. For that sake you need a formula."
> > If Jim Burns even had a modicum of brains, then he would have responded:
> >
> > "But you just said p/q ( "real number") is the process definition (formula)", which shows that you contradict yourself all the time.
> p/q is not a formula defining an individual fraction, like 1/3, but defining the general fraction.

Of course p/q is not a formula - p/q represents the general NAME of a number.

If p/q is not a "formula" as you call it, then neither is 1/3 because p/q is how 1/3 is represented.

You're silly and I am wasting my time with you.

>
> Regards, WM

On Tuesday, 29 March 2022 at 08:35:32 UTC+3, Eram semper recta wrote:
> On Monday, 28 March 2022 at 19:57:06 UTC+3, WM wrote:
> > Eram semper recta schrieb am Montag, 28. März 2022 um 06:48:59 UTC+2:
> > > On Monday, 28 March 2022 at 00:18:54 UTC+3, WM wrote:
> >
> > > > > They are complete well-formed numbers. They do not have to be represented in any particular radix system.
> > >
> > > > In principle they are instructions what to do: divide 1 by 3, if impossible multiply 1 by 10, write the result.
> > > Not at all.
> > 1/3 in words means "one divided by three".
> > > They are *NAMES* given to completed measures.
> > After the original meaning has been forgotten, they appear as names. Look, how many Millers are living in the USA? How many of them are producing flour?
> > > "Yes, we know how the set of real numbers is defined. But you cannot define all individuals. For that sake you need a formula."
> > > If Jim Burns even had a modicum of brains, then he would have responded:
> > >
> > > "But you just said p/q ( "real number") is the process definition (formula)", which shows that you contradict yourself all the time.
> > p/q is not a formula defining an individual fraction, like 1/3, but defining the general fraction.
> Of course p/q is not a formula - p/q represents the general NAME of a number.
>
> If p/q is not a "formula" as you call it, then neither is 1/3 because p/q is how 1/3 is represented.

p/q means number where p and q are symbols or place-holders for parts of the number (name).

p and q are NOT variables any more than say the x in x + 3 = 5 is a "variable". Context is important.

In the case of x+3=5, x is a CONSTANT, not a variable.

In the case of f(x)=x+3, x is a symbol or place-holder, not a variable. It's tempting to call x a variable in f(x)=x+3 but not actually correct. All the pairs (x, f(x)) are predefined and nothing actually "varies" or changes.

>
> You're silly and I am wasting my time with you.
>
> >
> > Regards, WM

On Monday, March 28, 2022 at 8:46:21 PM UTC-7, Jim Burns wrote:
> On 3/28/2022 1:03 PM, WM wrote:
> > Jim Burns schrieb
> > am Montag, 28. März 2022 um 00:40:14 UTC+2:
> >> On 3/27/2022 5:35 PM, WM wrote:
>
> >>> Yes, we know how the set of real numbers is defined.
> >>> But you cannot define all individuals.
> >>> For that sake you need a formula.
> >>
> >> Consider the possibility that we don't particularly
> >> want to define all individuals.
> >
> > If you want to map every individual
> > then you must define every indivdual.
> We describe the objects we intend to study.
>
> Here, we intend to study a geometric line.
> We describe it as having the _Archimedean property_
> and _Dedekind completeness_
>
> If your dark realm does not satisfy the description,
> we haven't forgotten it,
> it's not part of what we intend to study.
>
> One consequence of a line having the Archimedean property
> and Dedekind completeness is that there is no less than one
> and no more than one point which is in all intervals
> represented by an infinite digit sequence.
>
> That's
> no less than one point, definable or not definable,
> no more than one point, definable or not definable.
>
> Your argument is that
> Cantor is wrong because
> his system implies undefinable points,
> so, if his system is used,
> there will be points undefinable.
>
> Perhaps I missed it, but
> you've never answered the question
> "and then what?"

The Archimedean property is under-defined,
on the one sense it's granular, in the fine,
in the other it's fine, in the granular.

(In points and their topologies' neighborhoods.)

I define real numbers at least three ways:
integer part / non-integer part
non-integer part is between zero and one
fractions and irrationals
complete ordered field, algebraics
rationals and irrationals
signal field, uniform

All these numbers also have their values in their lattice,
also, and all their real spaces.

In the lattice of all real spaces.

Yeah, I think most people use all there of these real numbers.

Or, it's regular in common situations how they do.

There are models of each Hardy's real points as these.

Definition of continuity....

Then there is "now what".

On Monday, March 28, 2022 at 7:29:53 PM UTC+2, sergio wrote:
> On 3/28/2022 12:03 PM, WM wrote:
> > Jim Burns schrieb am Montag, 28. März 2022 um 00:40:14 UTC+2:
> > >
> > > Consider the possibility that we don't particularly want to define all individuals.
> > >
> > If you want to map every individual then you must define every indivdual.
> >
> does that apply to numbers ? or are you still Out Of Math (OOM)?

Well, he's never managed to actually DO math, since he's still defining each and every natural number, one after the other.