Deeper than primes

[doronshadmi]

As usual you simply want it both ways, to assert that it has “no sum” yet draw a conclusion “smaller than the value of the given limit” based on your simple assumption of a sum “smaller than the value of the given limit”

Sum is a result of a finite addition that reaches the value of a given limit, by finitely many steps.

Proportion is a result of an infinite addition that approaches the value of a given limit, by infinitely many steps.

“smaller than the value of the given limit” is not a sum, but it is a proportion, which is possible exactly because the added values > 0 have no sum (they do not reach the limit, exactly because we deal with infinitely many elements).

As for the Archimedean "proof":

What I (or more specifically Archimedes) did was multiply the series 1/2+1/4+1/8+1/16… by 2 meaning just adding it to itself.

What you are doing is this:

So as you see, nothing is changed, in both cases the added values only approach the limit (whether it is 1 or 2).

 

[doronshadmi]

No, that post points out how you mishandle standard mathematics to get the wrong answer. That isn't what I asked you. We already know you calculate very little correctly.

I asked you why you do it wrong then blame us,

jsfisher, your community plays a false game, as clearly shown in http://www.internationalskeptics.com/forums/showpost.php?p=5662065&postcount=8844 , and this is the exact answer to your question.

 

[jsfisher]

jsfisher, your community plays a false game, as clearly shown in http://www.internationalskeptics.com/forums/showpost.php?p=5662065&postcount=8844 , and this is the exact answer to your question.

Yes, your post does clearly show you get it wrong. No question about that. None at all. Again, my question was why do you blame us for your inability to follow simple rules?

 

[doronshadmi]

Yes, your post does clearly show you get it wrong. No question about that. None at all. Again, my question was why do you blame us for your inability to follow simple rules?

Rules? you change the rules in the middle of the game so blame only yourself.

 

[jsfisher]

Rules? you change the rules in the middle of the game so blame only yourself.

You clearly don't like the rules, but they don't change, and certainly not in the middle of the game. As I stated before, it is solely a case of you being unable to follow them. Please don't blame us for your failings.

 

[The Man]

Sum is a result of a finite addition that reaches the value of a given limit, by finitely many steps.

Again your simple assumption was proven wrong for a convergent infinite series some 2,300 years ago.

Proportion is a result of an infinite addition that approaches the value of a given limit, by infinitely many steps.

No proportion is the result of being, well, proportional. Apparently you simply do not understand that reaching some limit from some other value (with other values in between) specifically requires approaching that limit. Again approaching and reaching are not mutually exclusive.

“smaller than the value of the given limit” is not a sum,

It is a statement about the sum that it is “smaller than the value of the given limit”

but it is a proportion,

That would be another statement about the sum, which in the convergent series being discussed is a fractional value. You do understand that 1 to 1 (1/1) is a proportion, don’t you?

which is possible exactly because the added values > 0 have no sum (they do not reach the limit, exactly because we deal with infinitely many elements).

Again your simple assumption was proven wrong for a convergent infinite series some 2,300 years ago.

As for the Archimedean "proof":




What you are doing is this:

[qimg]http://farm3.static.flickr.com/2436/4394565223_68d292444a_o.jpg[/qimg]

No Doron that is just what you are doing, and once again by using a circle (or closed curve) you are asserting the circumference to be complete (that’s what makes it a circle or closed curve).

So as you see, nothing is changed, in both cases the added values only approach the limit (whether it is 1 or 2).

I see your one circle has a circumference of one and the other a circumference of two, meaning the circumference has increased by 1.

Again the proof is based on the fundimetal relations of1/2+1/4+1/8+1/16… =1/2+1/4+1/8+1/16… (one of your favorites) and 1/2+1/4+1/8+1/16… + 1/2+1/4+1/8+1/16… – (1/2+1/4+1/8+1/16… ) = 1/2+1/4+1/8+1/16… . In order to refute the proof you are going to have to show one of those relations to be false. So Doron which one is it? Drawing circles that you simply assume to be incomplete with an infinite number of elements is no different than your simple (and proven false) assumption that an infinite convergent series has no sum. Again your simple assumptions do not constitute proof and they certainly do not constitute the refutation of a proof.

 

[doronshadmi]

Apparently you simply do not understand that reaching some limit from some other value (with other values in between) specifically requires approaching that limit.

Not if approaching is a permanent state, and this is exactly the permanent state of proportion, in the case of the infinite geometric series 0+1/2+1/4+1/8+… (or 0+1/1+1/2+1/4+…) as clearly shown in:

In order to reach the limit in one of those geometric series, the reaching segment has to be at least in the same size of the previous segment, as clearly shown the diagrams above. But then we have a finite amount of segments.

Since any arbitrary segment along the infinite geometric series above is smaller then any arbitrary previous segment, and "being smaller then any arbitrary previous segment" is a permanent state of the infinite geometric series above, then those series do not reaching the value of the limit.

In other words, Archimedes did not prove anything about the infinite geometric series 0+1/2+1/4+1/8+… (or 0+1/1+1/2+1/4+…).

 

[doronshadmi]

Yes, your post does clearly show you get it wrong. No question about that. None at all. Again, my question was why do you blame us for your inability to follow simple rules?

jsfisher, 1/∞ = 0 is defined but ∞*0 or ∞*(1/∞) are undefined, and this is done by your community.

Can you explain such inconsistency?

 

[jsfisher]

jsfisher, 1/∞ = 0 is defined but ∞*0 or ∞*(1/∞) are undefined, and this is done by your community.

More correctly, 1/∞ is defined to be 0, while neither 0*∞ nor ∞/∞ are defined. Why does this trouble you? Do you also have difficulty dealing with 3/0 as being undefined?

Can you explain such inconsistency?

You haven't yet pointed out any inconsistency.


Also, what does any of this have to do with your bogus claim?

 

[doronshadmi]

Also, what does any of this have to do with your bogus claim?

What any of your false game (1/∞ is defined to be 0 (written as 1/∞=0), but ∞*(1/∞) or ∞*0 of ∞*(1/∞)=∞*0 are undefined) has to do with real reasoning?

 

[jsfisher]

What any of your false game (1/∞ is defined to be 0 (written as 1/∞=0), but ∞*(1/∞) or ∞*0 of ∞*(1/∞)=∞*0 are undefined) has to do with real reasoning?

You continue to evade the question. You make a bogus claim, and you can't back it up. Again you prove yourself an incompetent liar, and, since you cannot see the reasoning behind an elementary extension to basic arithmetic, an incompetent mathematician.

 

[The Man]

Not if approaching is a permanent state, and this is exactly the permanent state of proportion, in the case of the infinite geometric series 0+1/2+1/4+1/8+… (or 0+1/1+1/2+1/4+…) as clearly shown in:
[qimg]http://farm3.static.flickr.com/2436/4394565223_68d292444a_o.jpg[/qimg]

It is not “a permanent state”, once again your simple assumption that it is “a permanent state” (meaning the series has “no sum”), was proven wrong for a convergent geometric series some 2,300 years ago.

In order to reach the limit in one of those geometric series, the reaching segment has to be at least in the same size of the previous segment, as clearly shown the diagrams above. But then we have a finite amount of segments.

Nope it just has to be the same size as the remaining gap when “we have a finite amount of segments”, but we are not talking about a finite number of segments. Again Doron you are still focusing on your own concept of a ‘last segment’ which again has no relevance in an infinite series (there is no definable ‘last segment’).

Again do some research on your own. Let’s take the series 1/8+1/16+1/32… which is just what remains of your original series after the partial sum of ½+1/4 is removed. In other words the series 1/8+1/16+1/32… is one quarter of the original series and exactly equal to the “last segment” of that ½+1/4 partial sum as well as the reamining gap to the limit. Do it for any partial sum of your original series and you will find it is always the case. The remaining amount of that partial sum from the limit is exactly equal to the resulting infinite series after that ‘last segment’ of that partial sum and the sum of that new infinite series also equals that ‘last segment’ of the partial sum. So what ever ‘last segment’ you chose of the infinite series (resulting in a partial sum) the remaining infinite series is equal to the remaining gap as well as that ‘last segment’. Simply put your subsequent argument fails for an infinite series just as a definition for a ‘last segment’ does.

Since any arbitrary segment along the series is smaller then any arbitrary previous segment, and "being smaller then any arbitrary previous segment" is a permanent state of an infinite series, then those series do not reaching the value of the limit.

Since your concept of a ‘last segment’ is inapplicable to the infinite series so too is the above restriction you assert from the misapplication of that concept to the infinite series. Is that your problem Doron just you misapplying a concept from a finite series (or partial sum) to an infinite series?



Doron the reason you make the assumption you do (that an infnite convergent series has “no sum”) is irrelevant to the fact that it has been proven wrong some 2,300 years ago. One could also make the assertion that with each additional segment the series has a sum (referred to as a partial sum) thus the infinite series should also have sum. Now you may claim, and you do, that such an infinite sum is less than the limit of that series. However those too are simply assumptions based on certain aspects of the series, that the infinite series has a sum and that it is less than the limit. If only there was a way to use the relationships of that series to prove if the infinite series has a sum or not and if it is equal to the limit or not. Oh wait, there is and it was found some 2,300 years ago.

In other words Archimedes did not prove anything about the infinite geometric series 0+1/2+1/4+1/8+… (or 0+1/1+1/2+1/4+…).

He proved your assumption that a convergent infinite series has no sum is just wrong and that we can find that sum based on the common ratio. Since your assertion that the two times series (1/1+1/2+1/4+1/8+1/16…) has “no sum” is based solely on your assumption that the series 1/2+1/4+1/8+1/16… has “no sum” and that assumption was proven wrong some 2,300 years ago, it just makes your assertion that the two times series (1/1+1/2+1/4+1/8+1/16…) has “no sum” twice as wrong. Your assertions still amount to you simply claiming your assumption is correct only because you assume it to be correct.

 

[doronshadmi]

Since any arbitrary segment along the series is smaller then any arbitrary previous segment, and "being smaller then any arbitrary previous segment" is a permanent state of an infinite series, then those series do not reaching the value of the limit.

Since your concept of a ‘last segment’ is inapplicable to the infinite series

Excuse me ??

Where exactly do you find a 'last segment' in the quote above, taken from my post?

Again Doron you are still focusing on your own concept of a ‘last segment’ which again has no relevance in an infinite series (there is no definable ‘last segment’).

Exactly The Man. And this is the exact reason of why a finite amount of segments reaches the limit, and an infinite amount of segments does not reach the limit (there is no definable ‘last segment’ exactly because "smaller" is an invariant property of an infinite amount of added and converge elements > 0).

However those too are simply assumptions based on certain aspects of the series, that the infinite series has a sum and that it is less than the limit.

The Man, you are the one who using an assumption taken form a finite series (which has a sum) and force it on an infinite series, that has no sum exactly because it inherently incomplete.

Archimedes did not prove anything about the infinite geometric series 0+1/2+1/4+1/8+… (or 0+1/1+1/2+1/4+…), and the reason for this is clearly shown in http://www.internationalskeptics.com/forums/showpost.php?p=5667833&postcount=8867.

Your limited reasoning simply can't grasp infinite interpolation where "smaller" is its invariant state, no matter
how infinitely many converge elements > 0 are added.

the remaining infinite series is equal to the remaining gap as well as that ‘last segment’.

You simply assert that the remaining infinite series has a sum (as if it was a finite series) and then you are using this wrong assumption to reach the value of the limit.

 

[doronshadmi]

You continue to evade the question. You make a bogus claim, and you can't back it up. Again you prove yourself an incompetent liar, and, since you cannot see the reasoning behind an elementary extension to basic arithmetic, an incompetent mathematician.

It is a valid claim, your school of thouht plays a false game (1/∞ is defined to be 0 (written as 1/∞=0), but ∞*(1/∞) or ∞*0 of ∞*(1/∞)=∞*0 are undefined).

 

[sympathic]

Excuse me ??

Where exactly do you find a 'last segment' in the quote above, taken from my post?




Exactly The Man. And this is the exact reason of why a finite amount of segments reaches the limit, and an infinite amount of segments does not reach the limit (there is no definable ‘last segment’ exactly because "smaller" is an invariant property of an infinite amount of added and converge elements > 0).



The Man, you are the one who using an assumption taken form a finite series (which has a sum) and force it on an infinite series, that has no sum exactly because it is infinite and therefore inherently incomplete.

Archimedes did not prove anything about the infinite geometric series 0+1/2+1/4+1/8+… (or 0+1/1+1/2+1/4+…), and the reason for this is clearly shown in http://www.internationalskeptics.com/forums/showpost.php?p=5667833&postcount=8867.

Your limited reasoning simply can't grasp infinite interpolation where "smaller" is its invariant state, no matter
how infinitely many converge elements > 0 are added.

http://school.maths.uwa.edu.au/~berwin/humour/invalid.proofs.html

Some of Doron's favorites:

1.19 Proof by picture
A more convincing form of proof by example. Combines well with proof by omission.
1.20 Proof by vehement assertion
It is useful to have some kind of authority in relation to the audience.

 

[The Man]

Excuse me ??

Where exactly do you find a 'last segment' in the quote above, taken from my post?

Excuse me ??

Where did I claim you used the phrase “'last segment” in that post?

Are you claiming that your “reaching segment” would not be the 'last segment' in that series for you?

Exactly The Man. And this is the exact reason of why a finite amount of segments reaches the limit, and an infinite amount of segments does not reach the limit (there is no definable ‘last segment’ exactly because "smaller" is an invariant property of an infinite amount of added and converge elements > 0).

Again demonstrating the invalidity of your 'last segment' or if you prefer “reaching segment” argument for an infinite series.

The Man, you are using an assumption, taken form a finite series (which has a sum) and force it on an infinite series, that has no sum exactly because it is infinite and therefore inherently incomplete.

Doron I told you those were assumptions just as your assumptions are. The difference is you’re the only using and depending upon any of those assumptions. No one need assume a convergent infinite series has a sum or not when it is quite easy to prove that indeed it does have a finite sum.

Archimedes did not prove anything about the infinite geometric series 0+1/2+1/4+1/8+… (or 0+1/1+1/2+1/4+…), and the reason for this is clearly shown in http://www.internationalskeptics.com/forums/showpost.php?p=5667833&postcount=8867.

Again he proved that the assumptions you base that post on are simply false.

Your limited reasoning simply can't grasp infinite interpolation where "smaller" its invariant state, no matter how infinitely many converge elements > 0 are added.

Doron your limited reasoning simply can’t grasp that you simply assuming something for whatever reason does not constitute proof and certainly does not refute the given proof. Again in order to refute the given proof you would have to show that one or both of the two basic mathematical relationships it is base on (X=X and X+X-X=X) are wrong. So again, which one is wrong for you?

 

[doronshadmi]

http://school.maths.uwa.edu.au/~berwin/humour/invalid.proofs.html

Some of Doron's favorites:

1.19 Proof by picture
A more convincing form of proof by example. Combines well with proof by omission.
1.20 Proof by vehement assertion
It is useful to have some kind of authority in relation to the audience.

More nonsense from a community of people that enable to deal with abstraction only by using a very limited and specific way.

In other words, this community of people dependents on specific tools, which actually are nothing but a reflection of their own limitations to deal with abstraction, which is actually independent on any specific tool.

 

[doronshadmi]

(X=X and X+X-X=X) are wrong. So again, which one is wrong for you?

X+X is the right incomplete series and X is the left incomplete series in the following diagram:

[qimg]http://farm3.static.flickr.com/2436/4394565223_68d292444a_o.jpg[/qimg]

Excuse me ??

Where did I claim you used the phrase “'last segment” in that post?

http://www.internationalskeptics.com/forums/showpost.php?p=5668104&postcount=8872 your 3th part of your reply.

 

[zooterkin]

More nonsense from a community of people that enable to deal with abstraction only by using a very limited and specific way.

In other words, this community of people dependents on specific tools, which actually are nothing but a reflection of their own limitations to deal with abstraction, which is actually independent on any specific tool.

Look around you, and especially straight ahead at your computer, Doron, the world you live in was built using these tools that you dismiss because you don't understand them.

Now, exactly what has ever been accomplished with OM?

 

[doronshadmi]

Look around you, and especially straight ahead at your computer, Doron, the world you live in was built using these tools that you dismiss because you don't understand them.

Now, exactly what has ever been accomplished with OM?

It was built by using only finite tools (no infinite extrapolation\interpolation).

I am talking about the technology beyond the finite, which starts by dealing with infinite extrapolation\interpolation, which are inherently incomplete.