Deeper than primes

[HatRack]

Do this

1 + 1/2 + 1/4 + 1/27 + 1/256 + 1/3125 + 1/46656 + ...

step by step, so I would understand exactly what you mean when you talk about limits and infinite series.

I'm going to assume you mean 1 + 1/4 + 1/27 + ... + 1/n^n + ..., because otherwise it is simply this series with 1/2 added. Now, if the sequence of partial sums of the sequence {1, 1/4, 1/27, ..., 1/n^n, ...} has a limit L, then

1 + 1/4 + 1/27 + 1/256 + 1/3125 + 1/46656 + ... = L

by definition of the sum of a convergent series. For completeness, the sequence of partial sums is {1, 5/4, 139/108, ...}. It appears to have a limit, so L does exist and the series has a sum, by definition.

For any series denoted by the sequence {a1, a2, a3, ...} in general, its sum is the limit of the sequence of partial sums {s1, s2, s3, ...} provided that limit exists, where s1 = a1, s2 = a1 + a2, s3 = a1 + a2 + a3, ...

If the sequence of partial sums does not have a limit, as in the case of 1 + 1/2 + 1/3 + 1/4 + ..., then the expression does not denote a number. But, if it does have a limit, as in the case of 1/2 + 1/4 + 1/8 + ..., then the expression does denote a number, the limit of the sequence of partial sums.

That is the definition of infinite sum I have always used, and it is the definition in every textbook I have seen containing a section on infinite series.

 

[doronshadmi]

The uncountably infinite still escapes you, doesn't it?

No jsfisher, it does not escape me.

The collection of all uncountably infinite points along a line is also a collection of distinct points, and being distinct is possible only if no points are adjacent, otherwise it is the same point.

 

[doronshadmi]

Your conclusion doesn't follow from propositions C and D both being true. All you can establish from that is that if a point X is different from point 1, then point X is different from point 1.

Wrong, you are still missing the notion of being distict.

jsfisher, The uncountably infinite still (which is also a collection of distinct points) escapes you.

The uncountably infinite still escapes you, doesn't it?

No jsfisher, it does not escape me, but it clearly escapes you.

The collection of all uncountably infinite points along a line is also a collection of distinct points, and being distinct is possible only if no points are adjacent, otherwise it is the same point (a singleton set) with uncountably infinite names.

You continue to conflate adjacency with completeness. The continuum doesn't work that way.

The continuum works as follows:

1-dimensional space is strictly above any given collection of distinct 0-dimensional spaces along it.

 

[jsfisher]

Your conclusion doesn't follow from propositions C and D both being true. All you can establish from that is that if a point X is different from point 1, then point X is different from point 1.

Few will be impressed by this.

You continue to conflate adjacency with completeness. The continuum doesn't work that way.

jsfisher, The uncountably infinite still (which is also a collection of distinct points) escapes you.

The uncountably infinite still escapes you, doesn't it?

No jsfisher, it does not escape me, but it clearly escapes you.

The collection of all uncountably infinite points along a line is also a collection of distinct points, and being distinct is possible only if no points are adjacent, otherwise it is the same point.

Why are you shouting so? No one is disagreeing with this trivia, and that is all it is, trivia.

The point of contention is your baseless leap to the false conclusion that a line is not completely covered by points.

 

[doronshadmi]

Why are you shouting so? No one is disagreeing with this trivia, and that is all it is, trivia.

The point of contention is your baseless leap to the false conclusion that a line is not completely covered by points.

Your conclusion doesn't follow from propositions C and D both being true. All you can establish from that is that if a point X is different from point 1, then point X is different from point 1.

Wrong again, you are still missing the notion of being distict.

jsfisher, The uncountably infinite still (which is also a collection of distinct points) escapes you.

The uncountably infinite still escapes you, doesn't it?

No jsfisher, it does not escape me, but it clearly escapes you.

The collection of all uncountably infinite points along a line is also a collection of distinct points, and being distinct is possible only if no points are adjacent, otherwise it is the same point (a singleton set) with uncountably infinite names.

You continue to conflate adjacency with completeness. The continuum doesn't work that way.

The continuum works as follows:

1-dimensional space is strictly above any given collection of distinct 0-dimensional spaces along it, whether they are uncountably infinite or not.

 

[doronshadmi]

The point of contention is your baseless leap to the false conclusion that a line is not completely covered by points.

jsfisher, your notion is closed under the concept of Collection.

From this limited notion, one can't get the notion of distinction, and without the notion of distinction one can't comprehend that 1-dimensional space is strictly above any given collection of distinct 0-dimensional spaces along it, whether they are uncountably infinite or not.

 

[HatRack]

Wrong again, you are still missing the notion of being distict.

Doron,

Mathematics is not about notions, it is about axioms, definitions, and theorems. And, of course, theorems are further categorized into propositions, lemmas, and corollaries as appropriate. The only "notion", or undefined term, that we have in analysis is that of a set or collection. That is why we have axioms, to give us explicit and precise properties of undefined terms, so that we can deduce theorems.

If you wish anyone to take you seriously, you must lay an axiomatic groundwork upon which to base all of your logical deductions. Furthermore, you must clearly and precisely define all of the terms you use. Otherwise, this is an argument over definitions, which gets us nowhere.

 

[epix]

I'm going to assume you mean 1 + 1/4 + 1/27 + ... + 1/n^n + ..., because otherwise it is simply this series with 1/2 added. Now, if the sequence of partial sums of the sequence {1, 1/4, 1/27, ..., 1/n^n, ...} has a limit L, then

1 + 1/4 + 1/27 + 1/256 + 1/3125 + 1/46656 + ... = L

by definition of the sum of a convergent series. For completeness, the sequence of partial sums is {1, 5/4, 139/108, ...}. It appears to have a limit, so L does exist and the series has a sum, by definition.

For any series denoted by the sequence {a1, a2, a3, ...} in general, its sum is the limit of the sequence of partial sums {s1, s2, s3, ...} provided that limit exists, where s1 = a1, s2 = a1 + a2, s3 = a1 + a2 + a3, ...

If the sequence of partial sums does not have a limit, as in the case of 1 + 1/2 + 1/3 + 1/4 + ..., then the expression does not denote a number. But, if it does have a limit, as in the case of 1/2 + 1/4 + 1/8 + ..., then the expression does denote a number, the limit of the sequence of partial sums.

That is the definition of infinite sum I have always used, and it is the definition in every textbook I have seen containing a section on infinite series.

Well, with all the theory in the toolbox, the L is still uknown. But the feel for the series can give you an idea that is a bit more deterministic:

1 + 1/4 + 1/27 + 1/256 + 1/3125 + 1/46656 + ... < 1000

You can't prove it the way it is stated, but the inequality doesn't seem to be a complete nonsense, right? And that's the point.

 

[doronshadmi]

Doron,

Mathematics is not about notions, it is about axioms, definitions, and theorems. And, of course, theorems are further categorized into propositions, lemmas, and corollaries as appropriate. The only "notion", or undefined term, that we have in analysis is that of a set or collection. That is why we have axioms, to give us explicit and precise properties of undefined terms, so that we can deduce theorems.

If you wish anyone to take you seriously, you must lay an axiomatic groundwork upon which to base all of your logical deductions. Furthermore, you must clearly and precisely define all of the terms you use. Otherwise, this is an argument over definitions, which gets us nowhere.

http://www.internationalskeptics.com/forums/showpost.php?p=6588019&postcount=12528

http://www.internationalskeptics.com/forums/showpost.php?p=6590272&postcount=12545

http://www.internationalskeptics.com/forums/showpost.php?p=6590303&postcount=12546

 

[epix]

Are you sure you wanted that second term, there?

Toss it. But it actually doesn't matter, coz this series is very difficult even without that intruder.

 

[jsfisher]

The continuum works as follows:

1-dimensional space is strictly above any given collection of distinct 0-dimensional spaces along it, whether they are uncountably infinite or not.

Ah! Your favorite proof method: Proof by assumption. You assert your conclusion, then the proof just flows from there. I see.

And yet, Mathematics is unaware of your proof. It continues, apparently doing the impossible. A line, a plane, any higher dimension for that matter, is completely covered by those pesky 0-dimensional points.

They are just everywhere. Any nowhere are they not.

 

[jsfisher]

Toss it. But it actually doesn't matter, coz this series is very difficult even without that intruder.

Well, it is trivial to show it converges. Not so easy to give a closed-form result for it. (I know a nice integral, though.) At least it has a name.

 

[epix]

I'd like to read something about "line completely covered by points," that comes from the traditional mathematics sources, but the query doesn't work:
http://www.google.com/search?source...S333US333&q=line+completely+covered+by+points

 

[doronshadmi]

Ah! Your favorite proof method: Proof by assumption. You assert your conclusion, then the proof just flows from there. I see.

And yet, Mathematics is unaware of your proof. It continues, apparently doing the impossible. A line, a plane, any higher dimension for that matter, is completely covered by those pesky 0-dimensional points.

They are just everywhere. Any nowhere are they not.

jsfisher, thank you for support my argument about you.

You are closed under the concept of Collection, and this closeness is your axiomatic state of mind.

Being closed under that axiomatic state of mind, one can't get the notion of Distinction, and without the notion of Distinction one can't comprehend that 1-dimensional space is strictly above any given collection of distinct 0-dimensional spaces along it, whether they are uncountably infinite or not.

 

[HatRack]

Well, with all the theory in the toolbox, the L is still uknown. But the feel for the series can give you an idea that is a bit more deterministic:

1 + 1/4 + 1/27 + 1/256 + 1/3125 + 1/46656 + ... < 1000

You can't prove it the way it is stated, but the inequality doesn't seem to be a complete nonsense, right? And that's the point.

The inequality "1 + 1/4 + 1/27 + 1/256 + 1/3125 + 1/46656 + ... < 1000" is no more nonsense the the inequality "1/2 + 1/4 + 1/8 + ... < 2", although I don't feel like verifying it.

We can prove that the limit of the sequence of partial sums {1/2, 3/4, 7/8, ..., 1 - 1/2^n, ...} is less than 2, so "1/2 + 1/4 + 1/8 + ... < 2" is valid. You cannot prove, however, that the limit of the sequence is less than 1. In fact, you can prove that it is equal to 1. Hence, "1/2 + 1/4 + 1/8 + ... = 1" is valid by definition, and "1/2 + 1/4 + 1/8 + ... < 1" is invalid by trichotomy.

Someone stated this particular limit formally a few pages back in Latex, I believe. It is easily verified by working backwards to find an N for a given e > 0 such that for all n > N we have |(1 - 1/2^n) - 1| < e.

It is true that every partial sum of the series "1/2 + 1/4 + ..." is less than 1, which is easily proven. It is not true that the total sum of the series is less than 1, since the total sum of the series is defined to be the limit of the sequence of partial sums, which is provably 1.

 

[jsfisher]

jsfisher, thank you for support my argument about you.

You are closed under the concept of Collection, and this closeness is your axiomatic state of mind.

Being closed under that axiomatic state of mind, one can't get the notion of Distinction, and without the notion of Distinction one can't comprehend that 1-dimensional space is strictly above any given collection of distinct 0-dimensional spaces along it, whether they are uncountably infinite or not.

As is your technique, you blame others for your failure. You again assert as fact your conclusion, yet you offer no basis for that assertion. The failure is and has always been yours.

 

[HatRack]

Uhh, I asked for axioms/definitions. No such thing is contained in any of those posts. Here's an example of a clear, precise definition from elementary algebra:

The absolute value of a number x, denoted |x|, is equal to x when x > 0 or when x = 0, and it is equal to -x when x < 0.

Notice how the definition uses clear, precise language, and how it relies only on concepts that have been well-established such as number, equality, and inequality. Surely you can formulate your theory within a clear, precise framework of logical deduction after all the countless hours you've spent talking about it on this thread.

 

[laca]

No jsfisher, it does not escape me.

The collection of all uncountably infinite points along a line is also a collection of distinct points, and being distinct is possible only if no points are adjacent, otherwise it is the same point.

Doron, please define adjacent point or stop using the term. Your previous attempt at a definition was laughable. Try again.

 

[doronshadmi]

Uhh, I asked for axioms/definitions. No such thing is contained in any of those posts. Here's an example of a clear, precise definition from elementary algebra:

The absolute value of a number x, denoted |x|, is equal to x when x > 0 or when x = 0, and it is equal to -x when x < 0.

Notice how the definition uses clear, precise language, and how it relies only on concepts that have been well-established such as number, equality, and inequality. Surely you can formulate your theory within a clear, precise framework of logical deduction after all the countless hours you've spent talking about it on this thread.

HatRack, please support in details your argument, which asserts that the following is not formulated within a clear, precise framework of logical deduction:

Theorem: The collection of all distinct points of [0,1] can't completely cover [0,1].

Proof:

0 ≤ x ≤ 1

x=1 is trivially true, because x=1 is a collection of a single distinct point, which completely covers itself.

Proposition A: "Point x and point 1 are distinct"

Proposition B: "The distance between point x and point 1 is 0"

Proposition C: "A AND B is a contradiction"

Proposition D: "A AND ~B is a tautology"

Since C AND D is true, then the collection of all distinct points of [0,1] can't completely cover [0,1].

Q.E.D

 

[doronshadmi]

Doron, please define adjacent point or stop using the term. Your previous attempt at a definition was laughable. Try again.

laca, we have here jsfisher.

He is a professional mathematician, and he uses "adjacent" in the following post http://www.internationalskeptics.com/forums/showpost.php?p=6585515&postcount=12497 .

Is you wish to get an answer from a professional mathematician about "adjacent" , then just ask him.

After all he is the one that first used "adjacent" in this thread.

In case the you have missed it, I reply to jsfisher (and not to you) by using the same word that he first used.