Deeper than primes - Continuation

[realpaladin]

Awesome! We're back to the 'using discrete mathematics' to play with continuous.

Stop mixing paradigms, it leads to accidents.

You are stating a problem in continuous mathematics, but persist it to be solved with discrete mathematics.

What you state is analogous to:

- Say X is all the droplets in a glass of water.
- How many grains of sand are contained in that glass?

And JSFisher is right, using subscript numbers do not suddenly transform the infinite amount of Real numbers into a segmented array of Integers.

 

[doronshadmi]

That's a matter of definition. No need to restate it.



This lacks clarity. Based on how the rest of your post goes, I assume you mean something more like:

Let P be the ordered sequence 0 < a < b < c < d < ... < z < 1 and X = { [0,a], [a,b], [b,c], [c,d], ..., [z,1] }.​

This works fine for finite sequences, but if you intend to extend P to be an infinite sequence, it doesn't work so well. There is no last element, z, you can reference by its index.

So that's one problem you will need to resolve. Another is that an infinite sequence could have the same cardinality as R. Later in your post you assumed it to be limited to |N|.



This does not follow from anything you posted. Moreover, it is unlikely you will be able to show equivalence of any interval to an order relationship.

Thank you jsfisher for your remarks.

Let's start by using some particular example, and try to develop it without loss of generality.

First there is [0,1]

Ok, let's put the infinite sequence 1/2, 1/4, 1/8, 1/16, ... along [0,1], by using the convergent series 1/2+1/4+1/8+1/16+...

Please correct me if I am wrong, but as I understand it |{1/2, 1/4, 1/8, 1/16, ...}| = |N|

I have tried to express this sequence in terms of set of closed intervals along [0,1], as follows:

X =
{
[x₁,x₂], (is [0.0,0.5])
[x₂,x₃], (is [0.5,0.75])
[x₃,x₄], (is [0.75,0.875])
[x₄,x₅], (is [0.875,0.9375])
[x₅,x₆], (is [0.9375,0.96875])
...
}

Since there is a bijection form X to {1/2, 1/4, 1/8, 1/16, ...}, then |X| = |N|

Any X member is a closed interval that includes its own |R| amount of unique R members.

There is [x₁,x₂] which has only one overlap, and the other closed interval which has only one overlap is [x_?,1].

It must be stressed that without [x_?,1] as a member of X, X members can't fully cover [0,1] (or I am wrong here and we don't need [x_?,1] in order to fully cover [0,1], because (for example) [x₁,x₂] can fully cover [0,1] (but in that case why do we define a limit in the first place, if ,for example, [x₁,x₂] proper sub closed interval actually fully covers [0,1] ?)) .

(Actually, what mathematical definition makes the different sizes between [0,1] and [x₁,x₂] even is they have the same number of members?)

So what is the index of [x_?,1]?

In order to get [1,1] x_? must be at least x_|R|, but it is impossible since X can't include |R| as one of its own indexes, because |X|=|N|.

x_? also can't be x_|N|, since |N| can't be one of the indexes within a set, which its cardinality = |N|, and as we have shown |X|= |N|.

So what is left is some finite index, such that x_? is some x_n, and we get [x_n,1].

But then X has only finitely many members, but it is impossible since there is a bijection from X to {1/2, 1/4, 1/8, 1/16, ...}.

So, can we conclude that X does not have the closed interval of the form [x_#,1] and as a result all X members can't fully cover [0,1] (or in other words 1/2+1/4+1/8+1/16+... < 1)?

-------------------

EDIT:

It must be stressed that finitely many closed intervals (where each one of them has |R| R members) can actually fully cover [0,1], for example:

X={[0,1]}

X={[0,x₁],[x₁,1]}={[x₁,1],[0,x₁]} (the order of the members is insignificant, whether |X| is some finite cardinality, or even if |X|=|N| (in case that X is an infinite set)).

X={[0,x₁],[x₁,x₂],[x₂,x₃],...,[x_n,1]} etc. … (again, the order of the members is insignificant, whether |X| is some finite cardinality, or even if |X|=|N| (in case that X is an infinite set)).

Etc. ...

-----------------

I know that Cantor set has |R| members , but is it also a convergent sequence?

 

[jsfisher]

Thank you jsfisher for your remarks.

You are welcome.

Let's start form some particular example, and try to develop it without loss of generality.

First there is [0,1]

Ok, let's put the infinite sequence 1/2, 1/4, 1/8, 1/16, ... along [0,1], by using the convergent series 1/2+1/4+1/8+1/16+...

Please correct me if I am wrong, but as I understand it |{1/2, 1/4, 1/8, 1/16, ...}| = |N|

Fine so far.

First, I have tried to express this sequence in terms of set of closed intervals along [0,1], as follows:

X =
{
[x₁,x₂], (is [0.0,0.5])
[x₂,x₃], (is [0.5,0.75])
[x₃,x₄], (is [0.75,0.875])
[x₄,x₅], (is [0.875,0.9375])
[x₅,x₆], (is [0.9375,0.96875])
...
}

By the way, generating a set of intervals from the sequence does not add anything other than an unnecessary redirection. If you want to keep it simple, stick to just the sequence.

You have also impressed an order on both the sequence and the set of intervals. You should be more explicit about that.

Finally, you should be explicit as to how the x_i values are generated from the sequence. I would assume you meant it to be something like x_i = Sum(s_j, j = 1 to i-1), where <s₁, s₂, s₃, ...> is the generator sequence.

Since there is a bijection form X to {1/2, 1/4, 1/8, 1/16, ...}, then |X| = |N|

Sure, and this is also why you could restrict yourself to just the sequence.

Any X member is a closed interval of the general form [x_n,x_n+1] that includes its own |R| amount of unique R members.

Your phrasing is a bit awkward, but ok.

There is [x₁,x₂] which has only one overlap,

Overlap? Are you trying to say the interval [x₁,x₂] has a point in common with only one other member of X? Be that as it may, why is this observation at all important?

and the other closed interval which has only one overlap is [x_?,1].

Really? Now which interval would that be? There is no last member of the sequence you used to generate X, so why are you now alleging a member of X that would require there to be a final member of the sequence?

It must be stressed that without [x_?,1] as a member of X, X members can't fully cover [0,1].

Stress it all you like, but keep in mind that the way you have constructed your set X, the point 1 is not "covered".

Remember me pointing out that generating the set of intervals from the sequence was an unnecessary step? It was. Now consider your sequence. Even though its elements are monotonically decreasing towards 0, 0 itself does not ever appear in the sequence.

Zero is the limit of your sequence, but not an element of it. One is the limit for your generator function, but [1,1] is not an element of the generated set.

So what is the index of [x_?,1]?

In order to get [1,1]...

Your construction omits [1,1], so why does its absence surprise you?

...
So we have proved that only finitely many closed intervals (where each one of them has |R| R members can actually fully cover [0,1]

Really? You proved that? What about Y = X union {[1,1]}. Isn't Y an infinite set? Do the members of Y not fully cover the interval, [0,1]?

...
I know that Cantor set has |R| members , but is it also a convergent sequence?

Huh? Are your referring to the construction sequence used to generate Cantor's Set? Yes, the sequence converges.

 

[doronshadmi]

Generally, as I get it, the number of R members along a given closed interval > 0, has no impact on its length.

jsfisher, I am sorry, I have edited my post during your reply.

So, if you wish, please refresh your screen in order to see the edited version.

Sorry again and thank you.

 

[jsfisher]

Generally, as I get it, the number of R members along a given closed interval > 0, has no impact on its length.

I don't know what you mean by this.

jsfisher, I am sorry, I have edited my post during your reply.

So, if you wish, please refresh your screen in order to see the edited version.

Sorry again and thank you.

Yes, I noticed the additions, but it really doesn't change things. Your interval set construction leaves out one point, [1,1].

 

[doronshadmi]

Really? You proved that? What about Y = X union {[1,1]}. Isn't Y an infinite set? Do the members of Y not fully cover the interval, [0,1]?

Sorry, maybe I have missed something.

How exactly [1,1] which is a single point, can fully cover [0,1], in terms of length?

(Also, as I get it, the number of R members along a given closed interval > 0, has no impact on its length, and please correct me if I am wrong, as I get it convergent sequence is about length (including length 0)).

 

[doronshadmi]

I don't know what you mean by this.

For example [0,0.5] and [0,1] have the same cardinality (|R|) but not the same length.

Yes, I noticed the additions, but it really doesn't change things. Your interval set construction leaves out one point, [1,1].

Ok, How you construct it such that [1,1] is included?

 

[jsfisher]

Sorry, maybe I have missed something.

How exactly [1,1] which is a single point, can fully cover [0,1], in terms of length?

Do you accept that the members of your set of intervals, X, cover [0,1)? Then X union [1,1] would cover [0,1], no?

(Also, as I get it, the number of R members along a given closed interval > 0, has no impact on its length, and please correct me if I am wrong, as I get it convergent sequence is about length (including length 0)).

Your wording is awkward and hard to follow. Are you trying to say that the length of two intervals may be different even though the "number" of points on each interval is the cardinality of the continuum?

 

[doronshadmi]

Do you accept that the members of your set of intervals, X, cover [0,1)? Then X union [1,1] would cover [0,1], no?

Yes, but please show how you construct [0,1] by the series 1/2+1/4+1/8+1/16+... , such that [1,1] is included?

Your wording is awkward and hard to follow. Are you trying to say that the length of two intervals may be different even though the "number" of points on each interval is the cardinality of the continuum?

Yes. So what mathematical definition makes the difference?

 

[doronshadmi]

Jsfisher,

Please explain how exactly

X =
{
[x1,x2], (is [0.0,0.5])
[x2,x3], (is [0.5,0.75])
[x3,x4], (is [0.75,0.875])
[x4,x5], (is [0.875,0.9375])
[x5,x6], (is [0.9375,0.96875])
...
}

is different than Y = {1/2, 1/4, 1/8, 1/16, ...} ?

Thank you.

 

[jsfisher]

Yes, but please show how you construct [0,1] by the series 1/2+1/4+1/8+1/16+... , such that [1,1] is included?

I already did. You just didn't like the construction. I used your set, X, which was in fact constructed from the original sequence then added the singleton, [1,1], to make the set Y.

It seems like you are asking me to show where 0.999... occurs in the sequence, 0.9, 0.99, 0.999, 0.9999, .... It doesn't.

Yes. So what mathematical definition makes the difference?

I'm not sure where you are trying to go with this. Length makes the difference. Two intervals can differ in length. If [a,b] is an interval, then b-a is its length. (The same measure works for open and half-open intervals, too.)

Length is one measure than can be applied to an interval. Cardinality of the points along the interval would be another (although considerably less interesting).

 

[jsfisher]

Jsfisher,

Please explain how exactly

X =
{
[x1,x2], (is [0.0,0.5])
[x2,x3], (is [0.5,0.75])
[x3,x4], (is [0.75,0.875])
[x4,x5], (is [0.875,0.9375])
[x5,x6], (is [0.9375,0.96875])
...
}

is different than Y = {1/2, 1/4, 1/8, 1/16, ...} ?

Thank you.

First, lest there be any confusion, that's not my set, Y.

With your sets X and Y as given (and assuming the natural ordering and the generator function discussed earlier), there is a simple relationship between the two sets, and so they are, in that sense, equivalent.

Equivalent does not mean identical. Clearly X and Y are different because their memberships are disjoint. E.g., 1/2 is an element of Y but not of X.

 

[doronshadmi]

I already did. You just didn't like the construction. I used your set, X, which was in fact constructed from the original sequence then added the singleton, [1,1], to make the set Y.

It seems like you are asking me to show where 0.999... occurs in the sequence, 0.9, 0.99, 0.999, 0.9999, .... It doesn't.

Thank you for your explanation.

I do not ask you to show where 0.999... occurs in the sequence, 0.9, 0.99, 0.999, 0.9999 .

I ask you to show where 0 occurs in the sequence 0.9, 0.09, 0.009, 0.0009, ..., or more precisely, in the series 0.9+0.09+0.009+0.0009+... ?

And even if 0 is added to the length of the series 0.9+0.09+0.009+0.0009+... , how this series has length 1 by adding length 0 to it?

As I understand it, adding 0 length to finite or infinite series, has no impact on its length, exactly as adding the closed interval [1,1] (which has the finite cardinality |1| = |{1}|) to a sequence of closed intervals (where each one of them has cardinality |R|) has no impact on cardinality |R| (after all |R|+|1| = |R| , by transfinite arithmetic).

So I still do not understand how {[0,1),[1,1]} (which has the finite cardinality |2|), and {[0,1)} (which has the finite cardinality |1|), has any impact on cardinality |R| (after all |R|+|2| = |R|+|1| = |R| , by transfinite arithmetic).

Please rigorously define it, in order to help me to understand it (whether I dislike it, or not).

I'm not sure where you are trying to go with this. Length makes the difference. Two intervals can differ in length. If [a,b] is an interval, then b-a is its length. (The same measure works for open and half-open intervals, too.)

Please rigorously define how [0,0.999...) (which has cardinality |R| and length 0.999...) is actually length 1 by adding 0 length to length 0.999... if there is no impact on length, from the point of view of cardinality of the points along the interval [0,0.999...) ?

After all you wrote:

Length is one measure than can be applied to an interval. Cardinality of the points along the interval would be another (although considerably less interesting).

Thank you.

 

[doronshadmi]

First, lest there be any confusion, that's not my set, Y.

Ok, thank you for the clarification.

With your sets X and Y as given (and assuming the natural ordering and the generator function discussed earlier), there is a simple relationship between the two sets, and so they are, in that sense, equivalent.

Equivalent does not mean identical. Clearly X and Y are different because their memberships are disjoint. E.g., 1/2 is an element of Y but not of X.

I do not understand this part.

Please rigorously explain the difference between [0,0.5] and 1/2 .

After all both of them are essentially the same mathematical thing (known as length) by simply using different notations (and even if you think that cardinality is "considerably less interesting" |{[0,0.5]}| = |{1/2}| = |1|).

Thank you.

 

[jsfisher]

Thank you for your explanation.

I do not ask you to show where 0.999... occurs in the sequence, 0.9, 0.99, 0.999, 0.9999 .

I ask you to show where 0 occurs in the sequence 0.9, 0.09, 0.009, 0.0009, ..., or more precisely, in the series 0.9+0.09+0.009+0.0009+... ?

No where does 0 appear in the sequence. As for the series, I assume you meant 1 and not 0. The series 0.9 + 0.09 + 0.009 + ... is a single value, and that value happens to be 1.

And even if 0 is added to the length of the series 0.9+0.09+0.009+0.0009+... , how this series has length 1 by adding length 0 to it?

Series don't have length. Internals have length. Series have values (well, convergent series have values).

As I understand it, adding 0 length to finite or infinite series, has no impact on its length

Again, series do not have length. Intervals have length. For example, [0, 1) is a half-open interval of length 1. Adding [1, 1] to it yields the closed interval [0, 1], also of length 1.

...exactly as adding the closed interval [1,1] (which has the finite cardinality |1| = |{1}|) to a sequence of closed intervals (where each one of them has cardinality |R|) has no impact on cardinality |R| (after all |R|+|1| = |R| , by transfinite arithmetic).

So I still do not understand how {[0,1),[1,1]} (which has the finite cardinality |2|), and {[0,1)} (which has the finite cardinality |1|), has any impact on cardinality |R| (after all |R|+|2| = |R|+|1| = |R| , by transfinite arithmetic).

Why are you introducing the set { [0, 1), [1, 1] } into the discussion? Before we had only the set X, and I added the set Y (which was X union [1, 1]).

On the other hand, if you were to ask what points are "covered" by the elements of X, then it is all the points on the half-open interval, [0, 1). That is not a set; it is an interval. Similarly, the elements of Y cover [0, 1]. Again, an interval, not a set.

Please rigorously define it, in order to help me to understand it (whether I dislike it, or not).

Please rigorously define how [0,0.999...) (which has cardinality |R| and length 0.999...) is actually length 1 by adding 0 length to length 0.999... if there is no impact on length, from the point of view of cardinality of the points along the interval [0,0.999...) ?

Circling back to a much earlier observation, your conversion to intervals adds nothing to the discussion but an unnecessary layer of indirection. Your question about interval length is equivalent to the more direct question about 0.999... being equal to 1.

Remember that 0.999... is nothing more than an infinite series in disguise. The value for that series is taken from the limit of its finite counterparts, but we have been through all this before. You didn't like limits then, and I am guessing nothing has changed, so I see no reason to repeat the exercise.

On the other hand, were you able to demonstrate some inconsistency or contradiction arising from the definition of limits or from 0.999... = 1, that might be worth discussing. The unreasoned denial of before, though, not so much.

 

[jsfisher]

Ok, thank you for the clarification.


I do not understand this part.

Please rigorously explain the difference between [0,0.5] and 1/2 .

The former is an interval along the real line consisting of all the real numbers between 0 and 0.5, inclusive. The latter is simply the value, 0.5. At best one point if being interpreted in that context.

After all both of them are essentially the same mathematical thing (known as length) by simply using different notations (and even if you think that cardinality is "considerably less interesting" |{[0,0.5]}| = |{1/2}| = |1|).

As pointed out in a prior post, 1/2 does not have length. It is a value. On the other hand, [0, 0.5], does have length (the length is 0.5).

And why did you suddenly jump to sets, viz. {[0, 0.5]} and {1/2}? {1/2} and 1/2 are completely different things, just as [0, 1/2] and 1/2 are completely different things.

 

[doronshadmi]

jsfisher, thank you for the last two posts.

This is what I understand according to them:

A value is an abstract mathematical expression that its meaning is given according to a given context.

For example, 1 is an abstract and general mathematical expression, where one of its possible meanings is understood in terms of length, for example, the expression [1,1] is one of infinitely many possible ways in order to define length 0, where 0 is another example of an abstract and general mathematical expression.

What is called the real line, is a collection of such abstract and general mathematical expression.

One of the possible meanings of the real line is defined in terms of length, and one carefully has to distinguish between the real-line as an abstract and general mathematical expression, and one of its possible meanings (where length is an example of some particular meaning.

Please write your remarks to this post.

 

[jsfisher]

jsfisher, thank you for the last two posts.

This is what I understand according to them:

A value is an abstract mathematical expression that its meaning is given according to a given context.

Well, abstract concepts. Numbers are abstract concepts. Context can give them significance.

For example, 1 is an abstract and general mathematical expression, where one of its possible meanings is understood in terms of length, for example, the expression [1,1] is one of infinitely many possible ways in order to define length 0, where 0 is another example of an abstract and general mathematical expression.

Zero and 1 are numbers. Something can have a length of 1 (or 0), but I am less comfortable declaring 1, itself, as a length. It is a bit of a hair split, but sometimes that is important.

What is called the real line, is a collection of such abstract and general mathematical expression.

Numbers. There are many ways to express a given number. The real number line has an additional property that the numbers along the line are ordered.

One of the possible meanings of the real line is defined in terms of length, and one carefully has to distinguish between the real-line as an abstract and general mathematical expression, and one of its possible meanings (where length is an example of some particular meaning.

The real numbers, the real number line, and intervals along the real number line are all abstract concepts. So is length that one might attribute to an interval.

Remember, too, expressions are just that, ways of expressing things. Numbers, the real number line, and intervals along the real number line are all independent of how they might be expressed. Sixteen, 8+8, and 16.000 are all expressions for the same number.

 

[doronshadmi]

jsfisher, is the abstract concept length depends on the abstract concept number, but not vice versa (which means that there is an hierarchy of dependency, where number is more fundamental than length) ?

 

[doronshadmi]

On the other hand, were you able to demonstrate some inconsistency or contradiction arising from the definition of limits or from 0.999... = 1, that might be worth discussing.

{0.9, 0.09, 0.009, 0.0009, ...} is a one abstract mathematical object that has |N| members.

The series 0.9+0.09+0.009+0.0009+ ... is a one abstract mathematical object of convergent sequence of |N| values.

My question is this:

How a one abstract mathematical object of convergent sequence of |N| values is equal to a given value on the real line (known as the limit of that convergent sequence of |N| values) if this given value actually can be reached only by a convergent sequence of at least |R| values?