Cont: Deeper than primes - Continuation 2

[jsfisher]

Define convergent numbers please.

Please see http://www.internationalskeptics.com/forums/showpost.php?p=10614697&postcount=298.

I think Little 10 Toes already had a good idea of where you used the term. He was hoping you might actually define it.

 

[Little 10 Toes]

b is the limit in the following definition:

Given two numbers a and b such that a < b AND given sets S and K that their members are used as convergent series to b, if K is a proper subset of S such that |K|<|S|, then the sum of K members < b.

So let's see if you can respond without going back and editing your original post.

Define convergent series. Define limit.

 

[doronshadmi]

See that part I rendered in bold type? That's a conclusion. You cannot define conclusions; you have to prove them.

Define convergent series. Define limit.

Already done in http://www.internationalskeptics.com/forums/showpost.php?p=10364025&postcount=277.

 

[jsfisher]

Already done in http://www.internationalskeptics.com/forums/showpost.php?p=10364025&postcount=277.

Really? Well, since it isn't apparent you defined any terms or proved anything in that post from December, perhaps as a courtesy to the readers of today you would be so kind as to make a new post in which you define 'convergent series' and define 'limit' as you are using the terms.

Nothing else; just the definitions. That way we won't miss them buried among other clutter.

You see, one of the problems is that given the meaning of 'convergent series' in Mathematics, having sets where "their members are used as convergent series" is gibberish. So, perhaps you could tell us what you really meant, then we can help translate it to something that isn't gibberish.

(Just so you know, I'm expecting 'sequence' will be an important addition in untangling your word salad.)

 

[doronshadmi]

You see, one of the problems is that given the meaning of 'convergent series' in Mathematics, having sets where "their members are used as convergent series" is gibberish.

Well, there is no problem to understand that there is a set that its members are used as the numbers of a given sequence, for example:

The members of set {0.09,0.009,0.9,0.0009...} (where order does not matter) are used as the numbers of sequence (0.9,0.09,0.009,0.0009,...) (where order matters) that its sum (also known as convergent series) approaches a given number, called the limit of that sequence.

In case that S={0.09,0.009,0.9,0.0009...} and K is a proper subset of S such that |K|<|S|, that convergent series (which is based on the added K members) < the limit of the sequence that its numbers are S members.

|K|<|S| is clearly shown in terms of the tower of power as given in http://www.internationalskeptics.com/forums/showpost.php?p=10364025&postcount=277.

So according to what is written above, there is no problem to understand the following:

b is the limit in the following definition:

Given two numbers a and b such that a < b AND given sets S and K that their members are used as convergent series to b, if K is a proper subset of S such that |K|<|S|, then the sum of K members < b.

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

If we need |N|+1=|N| trick in order to conclude that 0.9₁₀+0.09₁₀+0.009₁₀+...=1 (as clearly shown in http://www.internationalskeptics.com/forums/showpost.php?p=10364025&postcount=277) then it is obvious that the sum of K members < b.

 

[jsfisher]

Well, there is no problem to understand that there is a set that its members are used as the numbers of a given sequence

Ah! So after my hint and some quality time with Google, you are beginning to understand that 'sequence' may have been the term you wanted.

for example:

The members of set {0.09,0.009,0.9,0.0009...} (where order does not matter) are used as the numbers of sequence (0.9,0.09,0.009,0.0009,...) (where order matters)

So, why not just start with a sequence of numbers? Unordered then ordered sets do not enhance the discussion, only confuse it in how you use and misuse the terms.

For that matter, your whole set diversion suffers in that sequence elements need not be unique within the sequence.

...that its sum (also known as convergent series) approaches a given number, called the limit of that sequence.

Most sequences are divergent, but what of that. As for the rest, the convergent series does not "approach" a given number. The series is the sum (or, more correctly, the limit of the sequence of partial sums) of the sequence.

For a convergent series, the series is a single value.

In case that S={0.09,0.009,0.9,0.0009...} and K is a proper subset of S such that |K|<|S|, that convergent series (which is based on the added K members) < the limit of the sequence that its numbers are S members.

Since S does not uniquely determine a sequence, you example is of no value.

If you want a sequence at the base of all this, then make S a sequence. And make K a sub-sequence of S if you must.

Also, you'll need to make sense of "the limit of the sequence that its numbers are S members."

And then, when all that is resolved, you'll need to construct a proof for your conclusion. Proof, not mere assertion as you are wont to do.

Finally, you'll need to show why the result (assuming you can establish the result) is at all interesting. It doesn't look like it will be interesting.

 

[doronshadmi]

If you want a sequence at the base of all this, then make S a sequence. And make K a sub-sequence of S if you must.

Thank you for your advice.

Since order does not matter, let S and K be mutlisets.

Also, you'll need to make sense of "the limit of the sequence that its numbers are S members."

The discussed elements are numbers.

And then, when all that is resolved, you'll need to construct a proof for your conclusion. Proof, not mere assertion as you are wont to do.

If the |N|+1=|N| trick is understood, then you immediately understand that this is a definition.

Finally, you'll need to show why the result (assuming you can establish the result) is at all interesting. It doesn't look like it will be interesting.

It is not interesting if the non-composed aspect of the tower of power line is excluded.

If it is not excluded then one immediately understands the non-entropic nature of the discussed framework, which is essential for further development of living creatures like us.

 

[jsfisher]

If you want a sequence at the base of all this, then make S a sequence. And make K a sub-sequence of S if you must.

Thank you for your advice.

Since order does not matter, let S and K be mutlisets.

You are just digging a hole. Put the shovel down. Your now multi-sets still do not determine the sequence. Take the short cut: Make S and K sequences.

Also, you'll need to make sense of "the limit of the sequence that its numbers are S members."

The discussed elements are numbers.

No, the limit of a sequence is just one number, the limit. And at least for your favorite example, that limit of sequence you are trying so desperately to represent by S is 0.

And then, when all that is resolved, you'll need to construct a proof for your conclusion. Proof, not mere assertion as you are wont to do.

If the |N|+1=|N| trick is understood, then you immediately understand that this is a definition.

Trick? It is no trick. But if the connection is so obvious, you will have no difficulty pointing it out at the proper time.

Finally, you'll need to show why the result (assuming you can establish the result) is at all interesting. It doesn't look like it will be interesting.

It is not interesting if the non-composed aspect of the tower of power line is excluded.

If it is not excluded then one immediately understands the non-entropic nature of the discussed framework, which is essential for further development of living creatures like us.

I will make no attempt to decode that gibberish.

 

[doronshadmi]

You are just digging a whole. Put the shovel down. Your now multi-sets still do not determine the sequence. Take the short cut: Make S and K sequences.

If I wish to use also cardinality as a factor of the discussed subject, I can't avoid mutisets, unless length can equivalently be used here, what do you think?

No, the limit of a sequence is just one number, the limit. And at least for your favorite example, that limit of sequence you are trying so desperately to represent by S is 0.

"the limit of the sequence that its numbers are S members" is b and the added members of S (which are all numbers) are sum=b, and as a result gibberish is the best you can get.

Trick? It is no trick. But if the connection is so obvious, you will have no difficulty pointing it out at the proper time.

Already done very simply by using your "Y = X union {[1,1]}" as seen in http://www.internationalskeptics.com/forums/showpost.php?p=10364025&postcount=277.

I will make no attempt to decode that gibberish.

Of course you will not do it, after all your framework arbitrarily excludes the non-composed aspect of the tower of power line.

 

[jsfisher]

You are just digging a whole. Put the shovel down. Your now multi-sets still do not determine the sequence. Take the short cut: Make S and K sequences.

If I wish to use also cardinality as a factor of the discussed subject, I can't avoid mutisets, unless length can equivalently be used here, what do you think?

I think you need to keep in mind that you are focusing on sequences. You keep trying to force sets as a vehicle towards that aim, but they are unnecessary and inappropriate. If the properties you seem to demand cannot be expressed directly for sequences, then perhaps you should not be including sequences at all.

One thing you are not allowed to side-step, too, is that a sequence is equivalent to a function over a totally ordered, countable domain. Keep that in mind as you wave your hands at cardinality.

No, the limit of a sequence is just one number, the limit. And at least for your favorite example, that limit of sequence you are trying so desperately to represent by S is 0.

"the limit of the sequence that its numbers are S members" is b and the added members of S (which are all numbers) are sum=b, and as a result gibberish is the best you can get.

Yes. See, that's the trouble with gibberish: It fails to have meaning.

Trick? It is no trick. But if the connection is so obvious, you will have no difficulty pointing it out at the proper time.

Already done very simply by using your "Y = X union {[1,1]}" as seen in http://www.internationalskeptics.com/forums/showpost.php?p=10364025&postcount=277.

That in no way proves anything about "that convergent series (which is based on the added K members) < the limit of the sequence that its numbers are S members." In fact, it would be rather difficult to prove anything of the sort since your usage of terms and relationships between sets and sequences is still all muddled.

Once you can finalize all the stuff that leads up to your conclusion, only then can we begin to consider its proof.

I will make no attempt to decode that gibberish.

Of course you will not do it, after all your framework arbitrarily excludes the non-composed aspect of the tower of power line.

Additional word salad does not assist in decoding the original gibberish.

 

[Little 10 Toes]

Already done in http://www.internationalskeptics.com/forums/showpost.php?p=10364025&postcount=277.

No it wasn't. The word limit was used in a sentence. Once. There is no listing of the phrase "convergent series". If you are going to take the time to dig up a source, make sure it supports you.

Try again.

Define convergent series. Define limit.

 

[doronshadmi]

I think you need to keep in mind that you are focusing on sequences. You keep trying to force sets as a vehicle towards that aim, but they are unnecessary and inappropriate.

Some important corrections (which correct also http://www.internationalskeptics.com/forums/showpost.php?p=10364025&postcount=277):

1) The order of the added numbers (where all numbers are positive) does not matter.

2) Number 0 does not contribute anything to the sum of the added numbers, and so is the case of multiple added numbers of the same value, the fact that they have the same value also does not contribute any important thing to the sum of the added numbers.

3) According to (1) and (2) the added numbers into a given series are actually reduced into unique members of non-empty sets.

So we can omit the usage of sequences, and use only the unique members of sets that are all positive numbers.

By simplifying the framework into the sum of added positive unique numbers, I discovered that K (where |K|<|S|) is not a proper subset of S because K has at least one member which is not a member of S.

Here is an example:

{0.9₁₀, 0.09₁₀, 0.009₁₀, 0.001₁₀} is some set with cardinality |4| that its added members provide 1 by one-step, and it is obvious that 0.001₁₀ is not a member of {0.9₁₀, 0.09₁₀, 0.009₁₀, ...} with cardinality |N| that its added members provide 1 by one-step.

So |K|<|S| and K is not a proper subset of S, yet it does not change the validity of my argument, which is as follow:

K or S must have smallest numbers > 0, such that the smallest number of S < the smallest number of K (such that the number of places of the smallest number of K < the number of places of the smallest number of S) and without those smallest numbers > 0, the sum of S or K members < the given limit.

If |K| is finite cardinality, the number of places of the smallest number of K is finite, otherwise it is infinite according to the cardinality of the set.

So we actually do not need the comparison between K and S in terms of set and proper subset, yet for all cardinality, there is the smallest number > 0 that the number of its places = the cardinality of its set.

The invariant existence of such smallest number > 0, proves the inability to define the smallest number > 0 for all sets, and we can conclude that no collection of added numbers has the non-composed property of 1-dimensional space because this property is stronger than the cardinality of all sets that their members are added along it from number a up to b, where a<b.

 

[doronshadmi]

Some corrections to my previous post:

Instead of


"So we actually do not need the comparison between K and S in terms of set and proper subset, yet for all cardinality, there is the smallest number > 0 that the number of its places = the cardinality of its set."

it has to be

"So we actually do not need the comparison between K and S in terms of set and proper subset, yet for all cardinality, there is the smallest number > 0 that the number of its places at the right side of the dismal point is finite or infinite (and in case that it is infinite, the number of places is according to the cardinality of its set)."

For example:

Exactly as 0.001₁₀ has finite places at the right side of the dismal point in case of {0.001₁₀, 0.9₁₀, 0.09₁₀, 0.009₁₀}, so is the case about 0.000...1₁₀ in case of {0.000...1₁₀, 0.9₁₀, 0.09₁₀, 0.009₁₀, ...}.

If one has no problem to understand that b is the result of infinitely many added numbers, one also does not have problems to understand that there is a number of the form 0.000...1₁₀ that has infinity many places at the right side of the dismal point that is > 0, that actually enables the members of {0.000...1₁₀, 0.9₁₀, 0.09₁₀, 0.009₁₀, ...} to be added up to b (even if 0 is a member of some set, it is obvious that it does not contribute anything to get result b).

So, without of numbers of the form 0.000...1₁₀ the result of the added members of sets with cardinality |N| actually < b.

The same principle holds with sets that their cardinality > |N|, but in this case we can't symbolize their members, because any string of symbols is limited to cardinality |N|.


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

Instead of

"The invariant existence of such smallest number > 0, proves the inability to define the smallest number > 0 for all sets,


it has to be

"The invariant existence of such smallest number > 0 for each set, proves the inability to define the smallest number > 0 of all sets that their members are added up to b from a to b, where a<b,"

 

[doronshadmi]

The ".000..." is used as a |N| size place value keeper that is inaccessible to "...1" that is at |R| size.

The same principle holds among |R|<|P(R)|<|P(P(R))|< ... ad infinitum, where no number of added values is accessible to the non-composed property of the line in itself, which is beyond collections.

Also in the case of sets that their members converge to a given number, known as the limit, these members can't be symbolized beyond cardinality |N|.

Here is the standard definition (http://en.wikipedia.org/wiki/Sequence#Definition_of_convergence):

Let (a_n) be a sequence. In words, the sequence (a_n) is said to converge if there exists a number L such that no matter how close we want the a_n to be to L (say ε-close where ε > 0), we can find a natural number N such that all terms (a_N+1, a_N+2, ...) are further closer to L (within ε of L). [1] This is often written more compactly using symbols. For instance,

for all ε > 0, there exists a natural number N such that L−ε < a_n < L+ε for all n ≥ N.

In even more compact notation

The last one can be written as follows:

for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |a_n - L| < ε.

So |a_n - L| must be = 0 in order to conclude that the sum over a_n = L.

For example, where is the rigorous proof that the sum over (0.9₁₀, 0.09₁₀, 0.009₁₀,...) = L = 1?

 

[jsfisher]

Some important corrections (which correct also http://www.internationalskeptics.com/forums/showpost.php?p=10364025&postcount=277):.

Post #277 is from four months ago. We've all moved on since then, including you. You introduced something new with its own set of misused terminology and obtuse word constructions. Are you abandoning that new, or just saving it for a link four months from now?

That sequence of posts from four months ago got you nowhere you wanted to go. You cannot pretend now it did, nor can you expect anyone here to re-experience the failure from that far back so you can get nowhere with it once again.

If you'd like to start anew, well, then you need to start anew. No silly links to different flavors of word salad.

 

[doronshadmi]

The solution to this question is by constructing a new kind of number that is based on the mathematical fact that |R|>|N|.

In the case of (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) the complement of 0.999...₁₀ to 1 is the number 0.000...1₁₀, where the ".000..." part is used as a |N| size place value keeper that is inaccessible to "...1" that is at |R| size.

"...1" at |R| size actually closes the gap between 0.999...₁₀ and 1, and without it 0.999...₁₀ = 1 is no more than an assertion.

 

[jsfisher]

The solution to this question is by constructing a new kind of number that is based on the mathematical fact that |R|>|N|.

In the case of

Have you considered composing your posts in a text editor first? Proofread and edit them there, and only when you are finally happy with the result, copy it to a post here.

 

[doronshadmi]

Have you considered composing your posts in a text editor first? Proofread and edit them there, and only when you are finally happy with the result, copy it to a post here.

Please follow it from http://www.internationalskeptics.com/forums/showpost.php?p=10617284&postcount=314.

 

[doronshadmi]

Are you abandoning that new,

No, I improve it by omitting the "|N|+1=|N|" stuff and its wrong related conclusion, which according to it 0.999...₁₀ = 1 without using 0.000...1₁₀.

But now I have realized that it can't be done without adding 0.000...1₁₀ to 0.999...₁₀

 

[jsfisher]

The solution to this question

What question?

...is by constructing a new kind of number that is based on the mathematical fact that |R|>|N|.

Great. After you tell us the question, please show us this construction. Don't start jumping to conclusions just yet. Let's see the construction itself.