Deeper than primes

[doronshadmi]

Here, for example, the challenge was for you to show that X is 0 in the limit.

No jsfisher, now you demonstrate that you have no clue what challenge has to be used in order to understand the considered subject.

The real challenge here is to show that X – (2a+2b+2c+2d+...) = 0

I proved that X – (2a+2b+2c+2d+...) > 0 exactly because X is a constant length upon infinitely many scale levels, and since (2a+2b+2c+2d+...) is found as long as constant X length is found, then since X can’t be length 0, then (2a+2b+2c+2d+...) < X, exactly because X can’t be length 0.

You babble instead about the relationship between X and a convergent series.

In other words, you do not understand http://www.internationalskeptics.com/forums/showpost.php?p=5725971&postcount=9152 proof,
where X is a constant and accurate length > 0 upon infinitely many scale levels, and (2a+2b+2c+2d+..) definitely converges upon infinitely many scale levels.

you admit the series is convergent, but will separately argue the series doesn't converge.

No jsfisher, (2a+2b+2c+2d+..) definitely converges (each arbitrary given segment is smaller than any arbitrary previous segment, upon infinitely many scale levels) exactly because constant X accurate length > 0 is found upon infinitely many scale levels.

Here is this beauty, that your limited Limit-oriented reasoning simply can’t comprehend:

 

[The Man]

Not once and not again, you have no argument The Man.

When you have something else to say, I am all ears.

It is you once again you Doron that has no argument.

Once again, X is a constant and accurate length > 0 in http://www.internationalskeptics.com/forums/showpost.php?p=5725971&postcount=9152 proof, that is not changed upon infinitely many scale levels.

By claiming the segments of “X” have “no sum” “upon infinitely many scale levels” your are indeed claiming it has change and is no longer “a constant”.

 

[jsfisher]

If we deal with an infinite convergent series, then exactly because of the inseparable linkage between constant X AND (2a+2b+2c+2d+...) convergent series, X-(2a+2b+2c+2d+...) > 0, and as a result (2a+2b+2c+2d+...) < X.

I see your reading comprehension problem continues.

In an earlier post, you said you had rigorously shown (your words, not those of anyone else) that "by standard mathematics" that X > 0 and X = 0, which would be a contradiction. What you said is being challenged.

Not that you showed it, rigorously or otherwise, I will agree that X > 0 at all steps in your construction. However, I do not agree X = 0, ever, neither at any step nor in the limit; I do not agree "by standard mathematics" X = 0; nor did you ever show that it was, rigorously or otherwise.

Please, stop with these irrelevant asides about the difference between X and some infinite series. Your incorrect and ridiculous insistence that you've shown X must be > 0 and = 0 at the same time ("by standard mathematics") is what is being challenged right now. Please focus.

Please support your bogus claim that X must be 0.

 

[jsfisher]

No jsfisher, now you demonstrate that you have no clue what challenge has to be used in order to understand the considered subject.

The real challenge here is to show that X – (2a+2b+2c+2d+...) = 0

The facts, is per usual, don't agree with you. In an earlier post you said:

Limit is valid only if it can be reached, and it can be reached only by finitely many steps.

Force your obsolete reasoning about Limits and X > AND = 0, which is false.

Simple as that, and you can't do anything about this fact.

Note the emphasized part. You said that. Either retract it as yet another bogus doronism where you again demonstrate your dearth of mathematical knowledge, or support the statement with yet another bogus doronism where you again demonstrate your dearth of mathematical knowledge.

Like it or not, nothing with limits in mathematics gets to the conclusion X both is > 0 and = 0. In your construction, X is always > 0, even in the limit.

 

[doronshadmi]

By claiming the segments of “X” have “no sum” “upon infinitely many scale levels” your are indeed claiming it has change and is no longer “a constant”.

No The Man.

X is an accurate value, called a sum.

(2a+2b+2c+2d+...) is an inaccurate value < X, and it is called (and here comes a novel concept) a fog.

Fogs are only approach to a given sum (what is called by you a limit).

For example: pi is a sum, and it is used by Standard Math as the limit of fog 3.14...[base 10], where fog 3.14...[base 10] < pi (fog 3.14...[base 10] only approaches sum pi).

In order to reach pi, one simply "jumps" form any arbitrary chosen scale level straight to sum pi, but then fog 3.14...[base 10] < pi is not found anymore, and we get a sum, which is based on finitely many segments AND points that have sum pi.

This novel reasoning about infinite convergent series is clearly demonstrated in http://www.internationalskeptics.com/forums/showpost.php?p=5721761&postcount=9104.

In other words, sums are local numbers, and fogs are non-local numbers.

By OM the place value method is a fog if infinitely many scale levels are involved.

By Standard Math the place value method is a representation of a sum (fogs are not found under the Standard framework).

Since Standard Math paradigm does not deal with fogs, then X can't be but a sum.

By using the limited reasoning, one simply can't get X as a fog (called also a non-local number, under OM).

 

[doronshadmi]

Like it or not, nothing with limits in mathematics gets to the conclusion X both is > 0 and = 0. In your construction, X is always > 0, even in the limit.

There is no "even in the limit" since the infinitely many scale levels of my construction are found exactly because the infinitely many added values (2a+2b+2c+2d+…) only approach X value, or in other words:

(2a+2b+2c+2d+…) < X, where < is an invariant fact.

On the contrary by Standard Math (2a+2b+2c+2d+…) = X, and since by my construction (2a+2b+2c+2d+…) is found as long as X is found, then if
X-(2a+2b+2c+2d+…)=0 (according to Standard Math) then X must be > 0 AND = 0 if (2a+2b+2c+2d+…) (which is found as long as X > 0) actually reaches X (and then X must be also 0 if (2a+2b+2c+2d+…) is still found also if it reaches X).

It is about the time for you to understand the difference between a sum and a fog (http://www.internationalskeptics.com/forums/showpost.php?p=5734631&postcount=9165).

 

[jsfisher]

...<snipped confused tangent>...

On the contrary by Standard Math (2a+2b+2c+2c+…) = X

At the limit, yes.

...and since by my construction (2a+2b+2c+2c+…) is found as long as X is found

"Is found"? It wasn't lost. It wasn't missing. The infinite sum simply is.

...then if X-(2a+2b+2c+2c+…)=0 (according to Standard Math) then X must be > 0 AND = 0

Why? You've somehow jumped from an incorrect understanding of limits to a completely disconnected conclusion. X is a constant > 0 throughout. What nonsense are you claiming leads you to conclude that X must also be 0?

...if (2a+2b+2c+2c+…) (which is found as long as X > 0) actually reaches X (and then X must be also 0 if (2a+2b+2c+2c+…) is still found also if it reaches X).

Would this be that nonsense? You have already admitted "by standard math" that (2a+2b+2c+2c+…) = X. Your disagreement with standard math doesn't change that that is the "by standard math" conclusion. So, clearly if (2a+2b+2c+2c+…) = X "by standard math", then of necessity X - (2a+2b+2c+2c+…) must = 0 "by standard math".

Nothing in those statements gets you to the conclusion X must be > 0 and = 0 "by standard math", though. Everything remains consistent with just X>0. Nothing implies X=0 also.

Again, doron, you only succeeded at another of your proof by misconception.

 

[doronshadmi]

At the limit, yes.



"Is found"? It wasn't lost. It wasn't missing. The infinite sum simply is.



Why? You've somehow jumped from an incorrect understanding of limits to a completely disconnected conclusion. X is a constant > 0 throughout. What nonsense are you claiming leads you to conclude that X must also be 0?



Would this be that nonsense? You have already admitted "by standard math" that (2a+2b+2c+2c+…) = X. Your disagreement with standard math doesn't change that that is the "by standard math" conclusion. So, clearly if (2a+2b+2c+2c+…) = X "by standard math", then of necessity X - (2a+2b+2c+2c+…) must = 0 "by standard math".

Nothing in those statements gets you to the conclusion X must be > 0 and = 0 "by standard math", though. Everything remains consistent with just X>0. Nothing implies X=0 also.

Again, doron, you only succeeded at another of your proof by misconception.

Since the infinite convergent series is found (2a+2b+2c+2c+…) upon infinitely many scale levels, as long as constant X>0 is found upon infinitely many scale levels, then (2a+2b+2c+2c+…) is considered as an infinite convergent series as long as it does not reach the value of constant X>0.

This is an irresistible logical fact about the infinite convergent series (2a+2b+2c+2c+…).

Since Standard Math insists that (2a+2b+2c+2c+…) is both infinite convergent series AND also has the exact value of X>0, then X must be also 0 in order to let the infinite convergent series (2a+2b+2c+2c+…) to reach the value of constant X>0.

Standard Math simply does not realize that claim that ((2a+2b+2c+2c+…) is an infinite convergent series) AND ( (2a+2b+2c+2c+…)=X ) is equivalent to the claim that X > AND = 0.

The reason to this false is based exactly on the indistinguishability between local and accurate values (that are reached by finitely many steps) and non-local an inaccurate values (that only approach by infinitely many steps).


The proof without words supports OM on this fine subject, and does not support Standard Math on this fine subject:

You ignored http://www.internationalskeptics.com/forums/showpost.php?p=5734631&postcount=9165.

Why is that?

"Is found"? It wasn't lost. It wasn't missing. The infinite sum simply is.

Infinite AND sum is a logical contradiction.

 

[doronshadmi]

Here is a part of a paper that is based on the Standard notion of infinitely many elements (http://www.ucl.ac.uk/philosophy/academic-research/watling/papers/TheSumOfAnInfiniteSeries.pdf John Watling, The sum of an infinite series, Analysis 13 (1952 - 53), 39 – 46.) (another address:http://www.pdfqueen.com/html/aHR0cD...GVycy9UaGVTdW1PZkFuSW5maW5pdGVTZXJpZXMucGRm):

The mistake that the completion of an infinite sequence of acts is logically impossible might
also be explained in this way.

The requirement that however many acts have been done, more must remain, is thought to entail that more acts must always remain.

But when the requirement is put explicitly as “whatever finite number has been done, more must remain”, it is clear that it does not entail that after an infinite number have been done more must remain.

By this part we can clearly see that Standard Math forces the notion of an accurate value (that is found by each finite step of infinitely many steps) on a value that is the result of infinitely many steps, by using the wrong notion "after" (or "all") in order to support the wrong notion that an infinite series has an accurate value exactly as a finite series of accurete values has an accurate value.

This logical failure is a direct result of Standard Math, which is tuned to deal only with accurate values, no matter what illogical payment is done, in order to achieve this goal.

Here is another part from this article:

It is important to distinguish this argument from another that is similar to it: Each of the additions that form the sum of 1 + ½ + ¼ + . . . gives a sum of a finite number of terms, e.g. ¼ to 1 + ½ gives a sum of three terms; since nothing else is required to form the sum of the series than making all these additions, the sum of the series must be a sum of a finite number of terms.

If this argument were valid we should have proved that the infinite series did not possess one of its defining properties, a contradiction from which it would follow that the sum did not exist.

But the argument is fallacious; it does not follow from the fact that each addition produces a certain result that all the additions together produce that result.

Because each of the additions yields a sum of a finite number of terms it does not follow that when all the additions are made they give a sum of a finite number of terms.

This is like arguing that the sum of an infinite number of the terms of a series must be equal to one of the terms of the sequence of sums of finite numbers of terms.

Well, this is the whole point, an infinite number of the terms of a series do not have an accurate value (which is a property that only a finite number of the terms of a series has).

Again Standard Math forces accuracy on an infinite collection, no matter what illogical payment is done, in order to achieve this goal.

And another part of this article:

Now from the definition of a limit we proved that each addition of a term of a series of positive
terms brought the sum of the terms nearer to the limit. The addition of each term brings the sum nearer to the limit than the sum of the finite number of terms preceding that term.

Therefore the addition of all the terms, forming the sum of the series, gives a sum nearer to the limit than the sum of any finite number of terms.

Of course no single addition brings the sum nearer than every sum of a finite number of terms.

Now from the definition of a limit, given any positive number a sum of a finite number of terms may be found such that the absolute value of the difference between it and the limit is less than that positive number.

Therefore the absolute value of the difference between the sum and the limit is less than any positive number.

But if the difference between two fixed quantities is less than any positive number, then they are equal.

Therefore the sum of an infinite series, whose terms all have the same sign, is equal to the limit of the sequence of sums of its terms.

Look what Standard Math does (the bolded part is mark by me):

It claims that there is a complete collection of positive numbers > 0, by examine each positive number and show that its value > 0.

By doing this Standard Math forces the accuracy that is found by each finite value, on a collection of infinitely many accurate values.

But it does not mean that if each member of some collection has an accurate value, then the value of infinitely many accurate values must be itself an accurate (or fixed) value.

So the whole notion of the Limit concept is based on a wrong notion.

 

[dafydd]

So the whole notion of the Limit concept is based on a wrong notion.

You appear to be the only person in the world who believes that,doesn't that tell you something?

 

[sympathic]

You appear to be the only person in the world who believes that,doesn't that tell you something?

It does. It means that he is the Newton of math.

 

[The Man]

No The Man.

X is an accurate value, called a sum.

(2a+2b+2c+2d+...) is an inaccurate value < X, and it is called (and here comes a novel concept) a fog.

Once again Doron deliberately being inaccurate and inconsistent, as you are, is in no way a “novel concept”.

Are you now claiming that your ‘bended’ segments always add up to “X is an accurate value, called a sum” even when there are an infinite number of your ‘bended’ segments?

Fogs are only approach to a given sum (what is called by you a limit).

For example: pi is a sum, and it is used by Standard Math as the limit of fog 3.14...[base 10], where fog 3.14...[base 10] < pi (fog 3.14...[base 10] only approaches sum pi).

In order to reach pi, one simply "jumps" form any arbitrary chosen scale level straight to sum pi, but then fog 3.14...[base 10] < pi is not found anymore, and we get a sum, which is based on finitely many segments AND points that have sum pi.

This novel reasoning about infinite convergent series is clearly demonstrated in http://www.internationalskeptics.com/forums/showpost.php?p=5721761&postcount=9104.

In other words, sums are local numbers, and fogs are non-local numbers.

By OM the place value method is a fog if infinitely many scale levels are involved.

By Standard Math the place value method is a representation of a sum (fogs are not found under the Standard framework).

Since Standard Math paradigm does not deal with fogs, then X can't be but a sum.

By using the limited reasoning, one simply can't get X as a fog (called also a non-local number, under OM).

Again what you call “Standard Math” deals quite well and with inaccurate values like estimates, rounded values and even ranges of values including their sums. A fact you would be well aware of if you had ever actually used what you call “Standard Math”. The only “fog” continues to be the one you deliberately restrict yourself to when it comes to what you call “Standard Math”.

 

[jsfisher]

Since the infinite convergent series is found (2a+2b+2c+2c+…) upon infinitely many scale levels, as long as constant X>0 is found upon infinitely many scale levels, then (2a+2b+2c+2c+…) is considered as an infinite convergent series as long as it does not reach the value of constant X>0.

"Is found"? You continue with the verbal nonsense. What do you mean by "is found"? Also, (2a+2b+2c+...) isn't "found" on infinitely many scale levels, either. It is an infinite series that has a single, precise value of X.

Be that as it may, X is a constant throughout your construction. The limit of X as the generation number approaches infinity is whatever constant value X started with at Generation 0. It does not change.

If K_n is the n-th generation Koch curve, and L(K_n) is the length of that curve, and W(K_n) is its width, then the limit as n approaches infinity of L(K_n) is X and of W(K_n) is 0.

Are you disputing these specific claims?

This is an irresistible logical fact about the infinite convergent series (2a+2b+2c+2c+…).

In the confused and contradictory world of doronetics, perhaps. In real Mathematics, no.

Since Standard Math insists that (2a+2b+2c+2c+…) is both infinite convergent series AND also has the exact value of X>0, then X must be also 0 in order to let the infinite convergent series (2a+2b+2c+2c+…) to reach the value of constant X>0.

By what miracle of logic do you get to that "then X must be also 0" conclusion? X is greater than 0. (2a+2b+2c+...) = X. That's all real Mathematics has to say about it. The rest you just made up.

[quoted]Standard Math simply does not realize that claim that ((2a+2b+2c+2c+…) is an infinite convergent series) AND ( (2a+2b+2c+2c+…)=X ) is equivalent to the claim that X > AND = 0.[/quote]

And yet you cannot explain why that is the case.

 

[doronshadmi]

Are you now claiming that your ‘bended’ segments always add up to “X is an accurate value, called a sum” even when there are an infinite number of your ‘bended’ segments?

No The Man.

I am talking about an accurate X value that is found upon infinitely many scale levels, where each level has a finite amount of bends.

I show that there is a 1-1 correspondence between any scale level of the constant and accurate X value, and any one of the accurate values of the infinite convergent series (2a+2b+2c+2d+...).

Again, it does not mean that if each value of the infinite convergent series (2a+2b+2c+2d+...) is an accurate value, then the added result of infinitely many accurate values is itself an accurate value.

A set of oranges is not itself an orange.

Again what you call “Standard Math” deals quite well and with inaccurate values like estimates, rounded values and even ranges of values including their sums.

Again, rounded values are accurate values and also ranges of values that (by your own words) have sums, can’t be but accurate values.

Inaccurate values are not defined as some accurate values that do not reach another given accurate value (called, a limit by you).

An Inaccurate value is defined as permanently approaches value that can’t reach (can’t be) any given accurate value, whether the accurate value is called Limit, or not.

 

[zooterkin]

I am talking about an accurate X value that is found upon infinitely many scale levels, where each level has a finite amount of bends.

Are you quite sure?

 

[doronshadmi]

If K_n is the n-th generation Koch curve, and L(K_n) is the length of that curve, and W(K_n) is its width, then the limit as n approaches infinity of L(K_n) is X and of W(K_n) is 0.

Are you disputing these specific claims?

L(K_n) is an invariant constant upon infinitely many scale levels, where the amount of the bends at each given scale level is finite. And this is exactly the reason of why X has an accurate value, no matter how infinitely many scale levels are involved.

“Approaches infinity” is a wrong notion simply because “approaches” is an essential property of any complex which is considered as infinite, and such a complex is resulted by an inaccurate value exactly because “approaches” is its essential property.

By what miracle of logic do you get to that "then X must be also 0" conclusion? X is greater than 0. (2a+2b+2c+...) = X. That's all real Mathematics has to say about it. The rest you just made up.

No, I clearly support my arguments all along the way, but you simply ignore them ( for example, now you ignore http://www.internationalskeptics.com/forums/showpost.php?p=5735873&postcount=9169) because your framework can’t deal with the real nature of inaccurate values.

 

[doronshadmi]

Are you quite sure?

Take for example the set of natural numbers.

It is infinite, yet each member is finite.

 

[ddt]

Take for example the set of natural numbers.

It is infinite, yet each member is finite.

I thought you didn't do infinite sets.

 

[doronshadmi]

I thought you didn't do infinite sets.

An infinite set does not have an accurate size, very simple.

Please be aware of the fact that I did not say "the set of all natural numbers".

 

[sympathic]

An infinite set does not have an accurate size, very simple.

Please be aware of the fact that I did not say "the set of all natural numbers".

No, of course not. You just said "the set of natural numbers" and stated that it is infinite. What is the difference between the "set of natural numbers" and the "set of ALL natural numbers"?