Deeper than primes - Continuation

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jsfisher said:
Your response, coupled with a link to an irrelevant post, ...
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http://www.internationalskeptics.com/forums/showpost.php?p=10112631&postcount=4147 is absolutely relevant and perfectly demonstrates your hands waving communication style, which does exactly nothing in order to get posts like http://www.internationalskeptics.com/forums/showpost.php?p=10100311&postcount=4128.

jsfisher said:
There is no such notion within ZFC according to its axiomatic foundation.
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According to the current agreement about ZFC, which is totally depends on the current subjective level of thoughts among traditional mathematicians like you, the notion of set is not a tautological existence within ZFC, so?

Also the notion of term is not a tautological existence within WFF, according to the current agreement about WFF among traditional mathematicians like you, so?

You are simply doing exactly nothing in order deal with ∃x, where x is a placeholder for set, term etc. and ∃ is used for tautological existence, no matter what domain of discourse is used.
 
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doronshadmi said:
http://www.internationalskeptics.com/forums/showpost.php?p=10112631&postcount=4147 is absolutely relevant
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That might be true if you were to completely ignore what I posted and all of its surrounding context. I prefer to work from what was actually written.

[And now, for something completely different...]
According to the current agreement about ZFC
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It is not an agreement. It is an axiomatic system.

With such an error at the start, there is no point examining the rest. Conclusions drawn from false premises are invalid.
 
jsfisher said:
It is not an agreement. It is an axiomatic system.
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It is a good example of how your subjective level of thoughts can't comprehend that the current ZFC axiomatic system is derived from your subjective level of thoughts.

In other words, you are doing exactly nothing to get new notions about ZFC axiomatic system.

You are completely ignore what I posted and all of its surrounding context, exactly as you are doing right now about the following:

You are simply doing exactly nothing in order deal with ∃x, where x is a placeholder for set, term etc. and ∃ is used for tautological existence, no matter what domain of discourse is used.

In case of set, Traditional Mathematics does not deal with tautological existence, and as a result ∃x (where in the case of ZFC axiomatic system, x is the notion of set) is not a valid expression of its domain of discourse.
 
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Sigh...yet again another failure by Doron to step up to the plate and provide what he promised me dozens of pages ago in this thread. A simple, no-nonsense, no excuses, no buzzword, easy-to-understand summary paragraph using laymans terms that explains why his method works.

Howl in the wind all you want, Doron. I'll be over there -->
 
jsfisher said:
You have yet to show that. All well-formed formulae, by their very definition, can be constructed from a small set of rules. ∃x cannot be constructed from those rules, and so ∃x is not a well-formed formula.
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So you are using the notion of set in order to define the rules of WFF, which are used to define sets.

In other words, circular reasoning.

"Great" agreement among traditional mathematicians, isn't it jsfisher?

Let me help you, the small set of rules is a set of terms, where each term stands at the basis of WFF set of rules.

So no matter how it is called, in both WFF or ZFC systems you are simply doing exactly nothing in order deal with ∃x, where x is a placeholder for notions like set, term etc., where ∃ is used for tautological existence, no matter what domain of discourse is used.

Tautological existence is excluded from any domain of discourse according to the current agreement among traditional mathematicians like jsfisher.

As a result of this current agreement ∃x is not a valid mathematical expression in any domain of discourse among traditional mathematicians like jsfisher.

jsfisher said:
Yes. You seem to be beginning to understand.
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Moreover, where their agreements are challenged, they simply ignore it, whether this ignoring style is called 100% discount or any other fancy name.
 
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doronshadmi said:
So you are using the notion of set in order to define the rules of WFF, which are used to define sets.
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No, I'm using colloquial language to express an idea.

Are you really so starved for attention you are dredging up all these old posts? Shouldn't you be off opening a TM clinic to save the Middle East from mass destruction?
 
jsfisher said:
No, I'm using colloquial language to express an idea.
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You wrote "by their very definition" so your colloquial language maneuvers are not impressive.

jsfisher said:
Are you really so starved for attention you are dredging up all these old posts?
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Time does not change the fact that your colloquial language maneuvers are not impressive.

jsfisher said:
Shouldn't you be off opening a TM clinic to save the Middle East from mass destruction?
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Shouldn't you be aware of the fact that 100% discount of X is no more than a colloquial language that is equivalent to ignoring X?

jsfisher said:
It is not an agreement. It is an axiomatic system.
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Also shouldn't you be aware of the fact that any given axiomatic system is based on an agreement among people about unproved expressions that are hopefully consistent with each other?

Or maybe your "It is not an agreement. It is an axiomatic system." is another demonstration of your colloquial language skills, who knows.
 
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doronshadmi said:
Time does not change the fact that your colloquial language maneuvers are not impressive.
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Doron please don't take offense but after 10,000+ posts it appears that you have failed to impress the great majority of posters on this board.

There is a recurring pattern. Someone will throw up an objection or a question or a concern, and after a few posts you'll give up and say they simply can't "get it", for whatever reason.

The majority of people on this forum seem to be reasonable individuals who are genuinely interested in being educated. Hence the "E" in JREF.

If after 10000+ posts you have failed to impress the majority of posters here with your theory / theories, it's not really our fault. It's your fault. I've been a teacher before, and while I've had the occasional student who just couldn't handle organic chemistry, most of my students "got it" after significantly less than 10,000+ interactions.

It's a sign that you're a poor teacher and a poor communicator of your ideas, because people are rejecting understanding them. Either that, or your ideas are simply wrong, and people are rejecting them for that reason.

But it's not the students fault if the teacher has ample lecture time and the earnest student still can't "get it". It's the teacher's fault. Either his ideas are wrong and the students rebel against that, or the concepts are so inefficiently communicated that students become hopelessly mired in terminology and conflicting concepts.

It's possible to become a better teacher. PM me if you like, and I can give you some tips on communication skills that I've learned during my career. Everyone has to learn these things, you're not born with them. You also need to consider that your ideas could be, well, wrong. They might need revising.

It could be a combination of the two. But you're not getting anywhere currently, so unless you want to continue blaming others for what are ultimately your problems, change. Learn about education, and revise your ideas to address serious concerns that others point out to you.

Otherwise, why should anyone listen to anything you have to say? You've had close to a decade and thousands of chances to explain your theory, and it hasn't worked, my man.
 
PiedPiper said:
Doron please don't take offense but after 10,000+ posts it appears that you have failed to impress the great majority of posters on this board.
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Let's use baby steps in order to rebuild the communication between us.

I say: "Wars start and end in people's minds, so the solution is profoundly related to people's minds".

Do you agree/disagree about that?
 
doronshadmi said:
Let's use baby steps in order to rebuild the communication between us.

I say: "Wars start and end in people's minds, so the solution is profoundly related to people's minds".

Do you agree/disagree about that?
Click to expand...


Off-topic BS. Please prove your claims.
 
doronshadmi said:
I take it as an axiom (where axioms do not need proves).

If you disagree that "Wars start and end in people's minds" then air your detailed view about it.
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Just one reason is needed here: "Hunger/food".

Unless you were imprecise and meant "Wars in the current times." (well, you could call the Boko Haram activities "hunger, but disguised as religion").

But then we still have:

"Disasters", "Ethnicity", "Religion" and I'll save a couple of others since I already know you by heart.

Whilst the latter two could be combined as "Culture", they are not in people's minds. Only misinformed people would say something like that.

Culture is the pattern and personal security in which you grow up. Changing that is not changing people's minds, but changing their environment.

And I would love to see you smooth out the culture of the Inuit with that of the Aboriginals.

But AdMan is right. This is another tangent you created so you do not need to answer the questions.
 
Let's assume that [0,1] is completely covered by R members.

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

So here the draft of my idea:

[0,1] is a non-empty closed interval that includes all the real numbers between 0 and 1 (including 0 and 1).

Let X be the set of closed intervals of R members that are included in [0,1], as follows:

X={[0,x1],[x1,x2],[x2,x3],[x3,x4],[x4,x5],…}, where |X|=|X={[0,x1],[x1,x2],[x2,x3],[x3,x4],[x4,x5],…}|

Each member of set X includes |R| amount of (almost only) unique R members along [0,1] (it is “almost only unique ...” because one or two R members in any given X member, overlapping one or two R members in one or two some other X member(s)).

If X has a finite cardinality, then there is no problem to completely 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],...,[xn,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)).

But what if |X|=|N|?

In that case the closed interval of the form [x|N|,1] can’t be in the range of set X (|N| can’t be used as an index within set X, because |X|=|N|) and we can conclude that X={[0,x1],[x1,x2],[x2,x3],[x3,x4],[x4,x5],…}, where no X member can reach number 1 of the interval [0,1].

The closed interval of the form [x|N|,1] is equivalent to the mathematical expression |N|<|R|.

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

Ok, I'll be thankful for your remakes, corrections, questions, ideas.
 
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doronshadmi said:
...
So here the draft of my idea:

[0,1] is a non-empty closed interval that includes all the real numbers between 0 and 1 (including 0 and 1).
Click to expand...

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

Let X be the set of closed intervals of R members that are included in [0,1], as follows:

X={[0,x1],[x1,x2],[x2,x3],[x3,x4],[x4,x5],…}, where |X|=|X={[0,x1],[x1,x2],[x2,x3],[x3,x4],[x4,x5],…}|
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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|.

...
The closed interval of the form [x|N|,1] is equivalent to the mathematical expression |N|<|R|.
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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.
 
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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.
 
jsfisher said:
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.
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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 xn, and we get [xn,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],...,[xn,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?
 
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doronshadmi said:
Thank you jsfisher for your remarks.
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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|
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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])
...
}
Click to expand...

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 xi values are generated from the sequence. I would assume you meant it to be something like xi = Sum(sj, 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|
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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 [xn,xn+1] that includes its own |R| amount of unique R members.
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Your phrasing is a bit awkward, but ok.

There is [x1,x2] which has only one overlap,
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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].
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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].
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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]...
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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]
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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?
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Huh? Are your referring to the construction sequence used to generate Cantor's Set? Yes, the sequence converges.
 
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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.
 
doronshadmi said:
Generally, as I get it, the number of R members along a given closed interval > 0, has no impact on its length.
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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.
Click to expand...

Yes, I noticed the additions, but it really doesn't change things. Your interval set construction leaves out one point, [1,1].
 
jsfisher said:
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]?
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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 said:
Sorry, maybe I have missed something.

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

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)).
Click to expand...

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?
 
jsfisher said:
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?
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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?


jsfisher said:
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?
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Yes. So what mathematical definition makes the difference?
 
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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.
 
doronshadmi said:
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?
Click to expand...

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?
Click to expand...

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).
 
doronshadmi said:
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.
Click to expand...

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.
 
jsfisher said:
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.
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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).


jsfisher said:
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.)
Click to expand...
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:
jsfisher said:
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).
Click to expand...

Thank you.
 
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jsfisher said:
First, lest there be any confusion, that's not my set, Y.
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Ok, thank you for the clarification.

jsfisher said:
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.
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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.
 
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doronshadmi said:
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+... ?
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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?
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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
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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).
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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...) ?
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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.
 
doronshadmi said:
Ok, thank you for the clarification.


I do not understand this part.

Please rigorously explain the difference between [0,0.5] and 1/2 .
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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|).
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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.
 
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.
 
doronshadmi said:
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.
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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.
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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.
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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.
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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.
 
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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) ?
 
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jsfisher said:
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.
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{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?
 
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