Deeper than primes - Continuation

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zooterkin said:
So, is 2 a member of {2} ?
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No.

2 is membership level 0, where {2} is membership level 1.

In other words, a membership level of a given set is the union of lower membership levels under this set, which is higher than any membership level under this union.

By this reasoning a set is the union of its members, where no member is at the membership level of the given set.

So the axiom of foundation is not violated, Russell's paradox is solved, and the complexity of membership levels is not ignored anymore.
 
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doronshadmi said:
No.

2 is membership level 0, where {2} is membership level 1.

In other words, a membership level of a given set is the union of lower membership levels under this set, which is higher than any membership level under this union.

By this reasoning a set is the union of its members, where no member is at the membership level of the given set.

So the axiom of foundation is not violated, Russell's paradox is solved, and the complexity of membership levels is not ignored anymore.
Click to expand...

You don't appear to be answering the question I asked. I didn't ask if 2 was the same as {2}.

If 2 is not a member of {2}, then what are the members of {2}?
 
doronshadmi said:
No.

2 is membership level 0, where {2} is membership level 1.

In other words, a membership level of a given set is the union of lower membership levels under this set, which is higher than any membership level under this union.
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This makes no sense. As far as I can tell, membership level measures the deepest level of nesting of brackets, if you will permit me to describe it in terms of its representation. Thus, Level 0: no members. Level 1: at least one member. Level N: at least one member with at least one member ... (total of N iterations of "at least one member"). In fact, is this the same as the standard set hierarchy? Maybe. Anyway, {2} is level 1, as you claim, precisely because it has a member, namely 2. Thus, 2 is a member of {2}, unless you have a completely non-standard definition of 'membership'.

About your claim that a set is the union of all lower levels under it: By "level under a set" I understand you to mean the set of members of the set at that level. So level 1 under the set S is the set of all level 1 sets in S. Same for other levels. The set of levels under S forms a partition of S, and thus their union is in fact the set, though the set is not the union of its members. For {2,{3}}, level 0 = {2}, level 1= {{3}}. The union of these levels is {2,{3}}, the original set. But the union of members of {2,{3}} is {3}. Not the original set. I do not see how it is possible to deny this (well, unless you use a definition of 2 such that it has members, but you explicitly deny this).

What good does any of this do us? What advance do we make over traditional mathematics?
 
zooterkin said:
You don't appear to be answering the question I asked. I didn't ask if 2 was the same as {2}.

If 2 is not a member of {2}, then what are the members of {2}?
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You don't appear to understand my answer.

If 2 is not the same as {2} (exactly because of the differences in membership levels), then how membership "2" expresses the membership of "{2}"?

"2 is a member of A(={2} or {2,...})" expression does not mean that this expression holds also if only "2" expression is used.

In other words, "2 is a member of {2}" is a complicated way to say "{2}".
 
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doronshadmi said:
You don't appear to understand my answer.

If 2 is not the same as {2} (exactly because of the differences in membership levels), then how membership "2" expresses the membership of "{2}"?

"2 is a member of A(={2} or {2,...})" expression does not mean that this expression holds also if only "2" expression is used.
Click to expand...


Ah!! You gotta love the classics!

Doron, now tell them about the severed body parts.
 
BenjaminTR said:
This makes no sense. As far as I can tell, membership level measures the deepest level of nesting of brackets, if you will permit me to describe it in terms of its representation. Thus, Level 0: no members. Level 1: at least one member. Level N: at least one member with at least one member ... (total of N iterations of "at least one member"). In fact, is this the same as the standard set hierarchy? Maybe. Anyway, {2} is level 1, as you claim, precisely because it has a member, namely 2. Thus, 2 is a member of {2}, unless you have a completely non-standard definition of 'membership'.
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According to my reasoning, a set is "more than nothing" thing, where the level of its membership is higher than any one of its members.

So set {} has membership level 0 (if it is determined by its "members"), where membership level 0 is more then nothing (nothing in itself has no membership level at all, which is not the same as membership level 0 (exactly as a point (0-dimensional space) is more than nothing).

So the higher membership level of a given set (where any set is "more than nothing") is determined by the union of lower membership levels of its members, where this higher level of membership is exactly the property that enables the union of lower membership levels, in the first place.

In other words, a set is the union of its members (its members are united by its higher level of membership).

BenjaminTR said:
For {2,{3}}, level 0 = {2}, level 1= {{3}}. The union of these levels is {2,{3}}, the original set. But the union of members of {2,{3}} is {3}. Not the original set. I do not see how it is possible to deny this (well, unless you use a definition of 2 such that it has members, but you explicitly deny this).
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For {2,{3}}, level_0 = 2, level_1 = {3}. The union of these levels is {2,{3}}, the original set.

"{2}u{{3}}" means "the union of membership level 0 of 2 with membership level 1 of {3} by membership level 2 of {2,{3}}" (where in the following examples the outer "{" and "}" are taken in term of in-vitro (they are isolated from a wider environment)).

(What is known as ur-elements are objects with membership level 0, where no one of them are nothing in itself).

BenjaminTR said:
What good does any of this do us? What advance do we make over traditional mathematics?
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The axiom of foundation is not violated, Russell's paradox is solved, the complexity of membership levels is not ignored, and the actual power of infinity (notated the outer "{" and "}" and understood by using also visual_spatial brain skills) is beyond any amount of united objects with lower membership levels (it has to be stressed that the outer "{" and "}" can be taken in terms of in-vitro or in-vivo).
 
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ZF set theory avoids Russell's paradox, and it includes the axiom of foundation. Are you trying to offer an alternative? If so, why prefer it to ZF (or its extensions)?
 
BenjaminTR said:
ZF set theory avoids Russell's paradox, and it includes the axiom of foundation. Are you trying to offer an alternative? If so, why prefer it to ZF (or its extensions)?
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The actual power of infinity of a non-composed dimensional space > 0 can't be addressed by ZF(C) or any other framework that is closed under the concept of collection.

If the actual power of infinity of a non-composed dimensional space > 0 is known (by using also visual_spatial brain skills) one naturally avoids complications like proper classes etc., and provides the simple foundation for Mathematics, as explained in http://www.internationalskeptics.com/forums/showpost.php?p=9205396&postcount=2344, http://www.internationalskeptics.com/forums/showpost.php?p=9208897&postcount=2352, http://www.internationalskeptics.com/forums/showpost.php?p=9209538&postcount=2354 and in more details in http://www.scribd.com/doc/98276640/Umes.

EDIT: Organic Mathematics does not ignore the principles of Evolution, it is not counter-intuitive and yet agrees with infinity (in its stronger version, that is derived from using also visual_spatial brain skills), it does not excludes the mathematician as a significant factor of mathematical development, which leads to the possibility to find the linkage among Ethics (in terms of evolutionary scale) and logical reasoning.
 
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doronshadmi said:
You don't appear to understand my answer.

If 2 is not the same as {2} (exactly because of the differences in membership levels), then how membership "2" expresses the membership of "{2}"?

"2 is a member of A(={2} or {2,...})" expression does not mean that this expression holds also if only "2" expression is used.

In other words, "2 is a member of {2}" is a complicated way to say "{2}".
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"2 is a member of {2}" is a sentence; it is the sort of thing that can be true or false. "{2}" is a singular term. It purports to refer to an individual. By linguistic considerations alone, your claim that they are equivalent is falsified. Consider the infelicity of "2 is a member of 2 is a member of {2}". That is, unless you can provide a compelling alternate semantics for the expressions in question, this does not work. (I would not rule an alternate semantics out a priori, there have been some interesting proposals about the semantics of mathematical terms, but for now, this claim is pretty well dead.)

As for how "2" denotes the membership of {2}: "2" is a name that refers to a number. That number is the only member of {2}. Thus, "2" denotes the membership of {2}. I literally cannot fathom any problem with this.
 
BenjaminTR said:
"2 is a member of {2}" is a sentence; it is the sort of thing that can be true or false. "{2}" is a singular term.
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"2" is a singular term, which means that it is at membership level 0.

"{2}" is not a singular term because at least two things (two membership levels) are involved here, which are:

1) The level of a member.

2) The level of a set.

And this is exactly what is involved in a sentence like "2 is a member of {2}".

Once again, "{}" is "more than nothing" thing at (at least) membership level 0 (if it is taken in-vitro), which is more than no membership at all (nothing in itself), exactly as a point (0-dimensional space) is "more than nothing" thing.

In other words, "{}" or "2" are (by following your words) examples of "singular terms", where "{2}" is not an example of a singular term.
 
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doronshadmi said:
If 2 is not the same as {2} (exactly because of the differences in membership levels), then how membership "2" expresses the membership of "{2}"?
Click to expand...
If you'd care to rephrase that in English, I might have a go at answering it.
"2 is a member of A(={2} or {2,...})" expression does not mean that this expression holds also if only "2" expression is used.
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Again, I'm not at all sure what you're trying to say; you appear to be bringing in things that are not relevant.
In other words, "2 is a member of {2}" is a complicated way to say "{2}".
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Have you changed your mind? Are you now agreeing that 2 is a member of {2}?


I've chosen a simple case, so we can focus on just one thing at a time. Sorry if you're having trouble with that.
 
zooterkin said:
If you'd care to rephrase that in English, I might have a go at answering it.

Again, I'm not at all sure what you're trying to say; you appear to be bringing in things that are not relevant.

Have you changed your mind? Are you now agreeing that 2 is a member of {2}?


I've chosen a simple case, so we can focus on just one thing at a time. Sorry if you're having trouble with that.
Click to expand...
http://www.internationalskeptics.com/forums/showpost.php?p=9214419&postcount=2371, sorry if you're having trouble with that.
 
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I corrected YESthing definition, in order to avoid two different interpretations of the outer "{" and "}" symbols:

YESthing (notated by the outer "{" and "}" symbols) is that is above members' membership.

I also corrected NOthing definition in order to save its consistent linkage with YESthing definition:

NOthing (not notated by any symbol) is that is below members' membership.

It is in the new version of http://www.scribd.com/doc/98276640/Umes.
 
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doronshadmi said:
"2" is a singular term, which means that it is at membership level 0.

"{2}" is not a singular term because at least two things (two membership levels) are involved here, which are:

1) The level of a member.

2) The level of a set.

And this is exactly what is involved in a sentence like "2 is a member of {2}".

Once again, "{}" is "more than nothing" thing at (at least) membership level 0 (if it is taken in-vitro), which is more than no membership at all (nothing in itself), exactly as a point (0-dimensional space) is "more than nothing" thing.

In other words, "{}" or "2" are (by following your words) examples of "singular terms", where "{2}" is not an example of a singular term.
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Ok, I will respond, however belatedly.

Perhaps I should have defined this the first time around, but a singular term is just a term purporting to refer to an individual. So 'Jesus' and 'the first star to go supernova' are singular terms. This contrasts with 'dogs' which is a bare plural and 'run'. These do not refer to individuals. In these terms, '2' and '{2}' are both singular terms. One refers to an integer, the other refers to a set. As long as {2} is an individual set, '{2}' is a singular term. The thing it refers to has several properties, but it itself is one thing.

With that out of the way, the rest of your post is about the properties of the set. You appear to admit in item 1 of your list that {2} has at least one member. What is that member? It has to be 2. This follows from the naming conventions established. There is a convention that a list of entities enclosed in curly brackets is the name for the set containing all and only the entities in the list as members. So yes, {} is more than nothing, it is the set containing nothing, since the list enclosed in brackets is empty. A set is not nothing (well, nominalism, fictionalism, blah blah whatever). The set {2} is also more than nothing. It even has a member. That member is 2.
 
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BenjaminTR said:
Ok, I will respond, however belatedly.

Perhaps I should have defined this the first time around, but a singular term is just a term purporting to refer to an individual. So 'Jesus' and 'the first star to go supernova' are singular terms. This contrasts with 'dogs' which is a bare plural and 'run'. These do not refer to individuals. In these terms, '2' and '{2}' are both singular terms. One refers to an integer, the other refers to a set. As long as {2} is an individual set, '{2}' is a singular term. The thing it refers to has several properties, but it itself is one thing.

With that out of the way, the rest of your post is about the properties of the set. You appear to admit in item 1 of your list that {2} has at least one member. What is that member? It has to be 2. This follows from the naming conventions established. There is a convention that a list of entities enclosed in curly brackets is the name for the set containing all and only the entities in the list as members. So yes, {} is more than nothing, it is the set containing nothing, since the list enclosed in brackets is empty. A set is not nothing (well, nominalism, fictionalism, blah blah whatever). The set {2} is also more than nothing. It even has a member. That member is 2.
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Expression "{2}" has at least two levels:

1) The level of being a member of a given set (notated as "2").

2) The level being a set (notated by the outer "{" and "}").

An expression that has more than one level can't be considered as a singular term.

Expression "2" has at most one level (it is a singular term) and it is not necessarily a member of a given set, exactly as "{}" has at most one level (it is a singular term) and it is not necessarily a member of a given set.
 
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Once again, by defining membership in terms of levels, the atomic (non-composed) level is level 0, where elements of membership level 0 are not necessarily related to higher membership level.

By following this notion {}, which is some level 0 element, may or may not be a member of {{}}, which is a composition of more than one membership level.
 
The sum of the length of the semi-circles at any given level of the following diagram is r*(Pi/2) (where in this case r=1, but r can be any length > 0):

9034009738_f34252f8b6_o.jpg


It is obvious that since r*(Pi/2) > r and r*(Pi/2) is an invariant size among any amount of levels of semi-circles, then also no infinitely many levels of semi-circles (where each level has r*(Pi/2) length) are reducible into r length.

According to the Limit notion r*(Pi/2) is reducible into r, which is equivalent to the claim that r*(Pi/2) = r*1 (or (Pi/2) = 1, which leads to the wrong conclusion that Pi = 2).

For example, please look how one (which uses the Limit notion) colcludes that Pi = 2:

http://plus.maths.org/content/puzzle-page-61
 
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doronshadmi said:
The sum of the length of the semi-circles at any given level of the following diagram is r*(Pi/2) (where in this case r=1, but r can be any length > 0):

[qimg]http://farm4.staticflickr.com/3803/9034009738_f34252f8b6_o.jpg[/qimg]

It is obvious that since r*(Pi/2) > r and r*(Pi/2) is an invariant size among any amount of levels of semi-circles, then also no infinitely many levels of semi-circles (where each level has r*(Pi/2) length) are reducible into r length.

According to the Limit notion r*(Pi/2) is reducible into r, which is equivalent to the claim that r*(Pi/2) = r*1 (or (Pi/2) = 1, which leads to the wrong conclusion that Pi = 2).
Click to expand...

The only things demonstrated here are that you don't understand limits and your so-called direct perception leads to wrong answers.

For example, please look how one (which uses the Limit notion) colcludes that Pi = 2:

http://plus.maths.org/content/puzzle-page-61
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How marvelously classic Doron. You contradict yourself. Look more closely. It is a puzzle. The author deliberately did something wrong to get the wrong result. He did the same wrong thing you did. The only difference is that he knew it was wrong.

Maybe if you actually read the article at the link you posted, you'd see the author explains what he did wrong to make the puzzle.
 
In the case of half circles, the length at each level of semi-circles remains r*(Pi/2), the following diagram:

9034009738_f34252f8b6_o.jpg


This is not the case with Koch fractal, where each level is longer than the initial straight line, in the following diagram:

koch.gif


The person is wrongly based on a model which is equivalent to Koch fractal, which is irrelevant to the semi-circles diagram.
 
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doronshadmi said:
In the case of half circles, the length at each level of semi-circles remains r*(Pi/2), the following diagram:

[qimg]http://farm4.staticflickr.com/3803/9034009738_f34252f8b6_o.jpg[/qimg]
Click to expand...

No one is disputing this except you.

This is not the case with Koch fractal, where each level is longer than the initial straight line, in the following diagram:

[qimg]http://www2.southeastern.edu/Academics/Faculty/jbell/koch.gif[/qimg]
Click to expand...

No one is disputing this, either, except you.

The person is wrongly based on a model which is equivalent to Koch fractal, which is irrelevant to the semi-circles diagram.
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What person is basing what model? Are you still babbling on about the puzzle link you posted? That person, the author of the puzzle, was not wrong at all, nor did he base his argument on the Koch snowflake curve.
 
The person is wrong because he\she writes about something that is irrelevant to the semi-circles case. Here it is:

The lesson is, when it comes to things like length, don't be fooled by appearance. A curve may look like a straight line, but it may be so wriggly at a small scale that in fact it's a lot longer than the line. Even worse, a set may look like a decent curve that should have finite length, but may in fact be infinitely long. And what exactly do we mean by "length" anyway? For an example of a curve with infinite length, check out the Koch curve in Plus article Jackon's fractals. And to learn about notions of length read Plus article Measure for measure.
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( http://plus.maths.org/content/puzzle-page-65 )

On the other hand the person is right only if he\she climes that no level of semi-circles is reducible into the length of the diameter, even if there are infinitely many levels of semi-circles.

This is not the case about the persons that agree with Limits, because according to their reasoning (when d = circle's diameter) d*(Pi/2) is reducible into d*1, where d*1 is the size of the limit, and according to this wrong notion since (d*(Pi/2))/d=1*(Pi/2)=Pi/2 is reducible into (d*1)/d=1*1=1, then Pi=2.

The same wrong reasoning is shown also in the case of 0.999...10=1 (in this case 0.999...10 is extended into 1, where in the previous case (d*(Pi/2))/d=1*(Pi/2)=Pi/2 is reducible into (d*1)/d=1*1=1, where both cases are equivalently wrong).
 
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doronshadmi said:
the person is wrong, here it is:

( http://plus.maths.org/content/puzzle-page-65 )
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You do like to jump around a lot, don't you. In what what is the person wrong in what you have hilighted? And please respond with clear, concise, declarative statements.

On the other hand the person is right only if he\she climes that no level of semi-circles is reducible into the length of the diameter, even if there are infinitely many levels of semi-circles.

This is not the case about the persons that agree with Limits, because according to their reasoning (when d = circle's diameter) d*(Pi/2) is reducible into d*1, where d*1 is the size of the limit, and according to this wrong notion since (d*(Pi/2))/d=1*(Pi/2)=Pi/2 is reducible into (d*1)/d=1*1=1, then Pi=2.
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Is this just a person you made up or did you have someone in mind? Certainly the author of that puzzle made no such claim. I made no such claim. You, well, hard to say what you are ever claiming, but since you neither like nor understand limits, I doubt you made any such claim.

Who, then?
 
Let us understand the following diagram, and how it proves the fallacy of the Limit reasoning:

9053976881_5e4453c2f2_o.jpg


According to the Limit reasoning, the blue segment (which its length = d*1/2 + d*1/4 + d*1/8 + d*1/16 + d*1/32 + ...) is equal to the green segment with length d (where d > 0).

By the following diagram:

Expression "1/2" is equivalent to expression "0.12"
Expression "1/4" is equivalent to expression "0.012"
Expression "1/8" is equivalent to expression "0.0012"
Expression "1/16" is equivalent to expression "0.00012"
Expression "1/32" is equivalent to expression "0.000012"

etc. at infinitum.

It has to be noticed that the length of the blue segment (which its length = d*0.12 + d*0.012 + d*0.0012 + d*0.00012 + d*0.000012 + ...) = d (the length of the green segment) only if Pi/2 is reducible into d.

But it is clearly shown that no amount of levels of semi-circles (where each level has length of Pi/2) is reducible into the length of d.

Since the purple semi-circles (where under each one of them there is a length of the form d*x2 (where any given x < 1 and > 0)) belongs to the infinitely many levels of the semi-circles that are irreducible into d, then no long addition of the form d*0.12 + d*0.012 + d*0.0012 + d*0.00012 + d*0.000012 + ... is equal to d.

In the particular case, where d=1, no long addition of the form 1*0.12 + 1*0.012 + 1*0.0012 + 1*0.00012 + 1*0.000012 + ... (= 0.11111...) is equal to 1 (unless Pi/2 is reducible into 1, but then Pi=2, which is clearly false).

Moreover, the added finite values of (for example) the long addition 1*0.12 + 1*0.012 + 1*0.0012 + 1*0.00012 + 1*0.000012 + ... (= 0.11111...), are equivalent to the finite amount of semi-circles at each level of the infinitely many levels (that their invariant length is irreducible into 1 (if d=1)).
 
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doronshadmi said:
Let us understand the following diagram, and how it proves the fallacy of the Limit reasoning:

[qimg]http://farm3.staticflickr.com/2847/9053976881_5e4453c2f2_o.jpg[/qimg]

According to the Limit reasoning, the blue segment (which its length = d*1/2 + d*1/4 + d*1/8 + d*1/16 + d*1/32 + ...) is equal to the green segment with length d (where d > 0).

By the following diagram:

Expression "1/2" is equivalent to expression "0.12"
Expression "1/4" is equivalent to expression "0.012"
Expression "1/8" is equivalent to expression "0.0012"
Expression "1/16" is equivalent to expression "0.00012"
Expression "1/32" is equivalent to expression "0.000012"

etc. at infinitum.

It has to be noticed that the length of the blue segment (which its length = d*0.12 + d*0.012 + d*0.0012 + d*0.00012 + d*0.000012 + ...) = d (the length of the green segment) only if Pi/2 is reducible into d.

But it is clearly shown that no amount of levels of semi-circles (where each level has length of Pi/2) is reducible into the length of d.

Since the purple semi-circles (where under each one of them there is a length of the form d*x2 (where any given x < 1 and > 0)) belongs to the infinitely many levels of the semi-circles that are irreducible into d, then no long addition of the form d*0.12 + d*0.012 + d*0.0012 + d*0.00012 + d*0.000012 + ... is equal to d.

In the particular case, where d=1, no long addition of the form 1*0.12 + 1*0.012 + 1*0.0012 + 1*0.00012 + 1*0.000012 + ... (= 0.11111...) is equal to 1 (unless Pi/2 is reducible into 1, but then Pi=2, which is clearly false).

Moreover, the added finite values of (for example) the long addition 1*0.12 + 1*0.012 + 1*0.0012 + 1*0.00012 + 1*0.000012 + ... (= 0.11111...), are equivalent to the finite amount of semi-circles at each level of the infinitely many levels (that their invariant length is irreducible into 1 (if d=1)).
Click to expand...

Yes. When a sequence, for example, tends to a limit, generally each successive iteration or term added produces a result closer to the limit (or at least that's the trend of the series). In this example, each successive iteration (halving the diameter of the semi-circles), produces exactly the same result as the previous one.

If you look at the explanation:
The solution

What's wrong here? Well, it is definitely true that after n steps there will be 2n semi-circles, each of length 1/2n. It is also true that the diameter of the largest circle is in some sense a limit of the strings of semi-circles: by making n large enough, you can ensure that the nth string of semi-circles squeezes as closely to the diameter d as you like.

The mistake lies in the next step: the assumption that the lengths of the strings of semi-circles tend to the length of the diameter. This is false! What happens in fact is that while at each stage we replace the semi-circles involved by smaller ones, these also become more numerous, so that the replacement makes no difference at all to the overall length. Every string of semi-circles has length 1, since

2n×1/2n = 1,

for any n. We've simply replaced a number of big wriggles by a greater number of smaller wriggles without changing the length.
The lesson is, when it comes to things like length, don't be fooled by appearance. A curve may look like a straight line, but it may be so wriggly at a small scale that in fact it's a lot longer than the line. Even worse, a set may look like a decent curve that should have finite length, but may in fact be infinitely long.
Click to expand...

I suggest the highlighted warning is something you should heed.
 
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zooterkin said:
Yes. When a sequence, for example, tends to a limit, generally each successive iteration or term added produces a result closer to the limit (or at least that's the trend of the series). In this example, each successive iteration (halving the diameter of the semi-circles), produces exactly the same result as the previous one.
Click to expand...
Exactly, and this is the reason of why Pi/2 is irreducible into d length (where d is the limit) no matter how many levels of semi-circles (where each level has Pi/2 length) are involved.

zooterkin said:
If you look at the explanation:

...don't be fooled by appearance...


I suggest the highlighted warning is something you should heed.
Click to expand...
I suggest that you combine your visual_spatial AND verbal_symbolic brain skills in order to understand http://www.internationalskeptics.com/forums/showpost.php?p=9296954&postcount=2389.

Only then you will not be fooled by (verbal_symbolic-only OR visual_spatial-only) appearance.
 
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doronshadmi said:
Let us understand the following diagram, and how it proves the fallacy of the Limit reasoning:

[qimg]http://farm3.staticflickr.com/2847/9053976881_5e4453c2f2_o.jpg[/qimg]

According to the Limit reasoning, the blue segment (which its length = d*1/2 + d*1/4 + d*1/8 + d*1/16 + d*1/32 + ...) is equal to the green segment with length d (where d > 0).

By the following diagram:

Expression "1/2" is equivalent to expression "0.12"
Expression "1/4" is equivalent to expression "0.012"
Expression "1/8" is equivalent to expression "0.0012"
Expression "1/16" is equivalent to expression "0.00012"
Expression "1/32" is equivalent to expression "0.000012"

etc. at infinitum.

It has to be noticed that the length of the blue segment (which its length = d*0.12 + d*0.012 + d*0.0012 + d*0.00012 + d*0.000012 + ...) = d (the length of the green segment) only if Pi/2 is reducible into d.
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Thee's your problem, right there. Why would you write such a ridiculous thing? You have imagined a dependency that doesn't exist.

The limit of your blue path length is 1, and the limit of your violet path length is pi/2. Neither "reduces" to the other, neither singly nor by way of an infinite sequence or series, ever. The arc length of the semicircle is always pi/2 times its diameter, always.
 
We compare between the blue line and the green line, where the blue line is always < the green line, and this is exactly my argument about the fallacy of the Limit reasoning, as shown in http://www.internationalskeptics.com/forums/showpost.php?p=9296954&postcount=2389.

Here is the part that was misunderstood:

"Since the purple semi-circles (where under each one of them there is a length of the [blue] form d*x2 (where any given x < 1 and > 0)) belongs to the infinitely many levels of the semi-circles that are irreducible into d, then no long addition of the form d*0.12 + d*0.012 + d*0.0012 + d*0.00012 + d*0.000012 + ... is equal to d."

In the particular case, where d=1, no long addition of the [blue] form 1*0.12 + 1*0.012 + 1*0.0012 + 1*0.00012 + 1*0.000012 + ... (= 0.11111...2) is equal to 1 (unless Pi/2 is reducible into 1, but then Pi=2, which is clearly false)."
 
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doronshadmi said:
We compare between the blue line and the green line, where the blue line is always < the green line, and this is exactly my argument about the fallacy of the Limit reasoning, as shown in http://www.internationalskeptics.com/forums/showpost.php?p=9296954&postcount=2389.
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Assuming the green line segment includes its endpoints, it is not identical to the blue segment, which does not include the endpoint on the right. This does not mean they differ in length, however. They are the same length.

You seem to imply here that if every member of a series of real numbers is less than x, then the limit of the series is less than x. This is not true; if x is the limit this means that no matter how small a number epsilon you pick, you can always find a member of the series such that every later member of the series is less than epsilon away from x. From this definition it is provable that 1 is the limit as n approaches infinity of the series of sums 1/2 + 1/4 + ... 2-n. This is true even though every individual member of the series is less than 1. Consider: every finite integer is finite, but there are infinitely many of them. Infinity is very cool, and sets and series with infinitely many members have sometimes surprising properties.
 
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doronshadmi said:
We compare between the blue line and the green line, where the blue line is always < the green line, and this is exactly my argument about the fallacy of the Limit reasoning
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No, you didn't. You went off on some other tangent about "reducibility" of the arc length to the diameter. There was no argument you presented there, and there is no argument you are presenting now.

You don't understand limits. You don't like them, and you make ignorant assertions thinking you are proving them bogus.

If you want to demonstrate limits are faulty in some way, your personal incredulity doesn't do it. You will need to work from what "limit" actually means (you know, those delta and epsilon things), not simple wave your hands in disbelief over some blue and green lines.

...as shown in http://www.internationalskeptics.com/forums/showpost.php?p=9296954&postcount=2389.
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That would be where you showed you have trouble expressing anything resembling a cohesive argument. Coupled with your post cited here, together they show you cannot follow your own argument.
 
doronshadmi said:
Let us understand the following diagram, and how it proves the fallacy of the Limit reasoning:

[qimg]http://farm3.staticflickr.com/2847/9053976881_5e4453c2f2_o.jpg[/qimg]

According to the Limit reasoning, the blue segment (which its length = d*1/2 + d*1/4 + d*1/8 + d*1/16 + d*1/32 + ...) is equal to the green segment with length d (where d > 0).

[...]

It has to be noticed that the length of the blue segment (which its length = d*0.12 + d*0.012 + d*0.0012 + d*0.00012 + d*0.000012 + ...) = d (the length of the green segment) only if Pi/2 is reducible into d.

[...]
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I am thinking visually and spatially. I am even looking at the diagram. It is plain to see that no matter how many circles you pile on, or how big or small they get, the circumference will always be greater than the diameter. Thus, we can conclude, using verbal-symbolic-visual-spatial reasoning that the length of the blue segment equals d only if the length of the purple curve is greater than d. In fact, we can say more since we know the purple curve is built of half circumferences of circles with diameters summing to the blue line segment. Thus we know the blue segment has length d if and only if the purple one has length d*Pi/2. Your claim that the blue is length d only if the purple reduces to d, is demonstrably false, even using only visual reasoning, but especially using all the reasoning skills you promote.
 
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BenjaminTR said:
You seem to imply here that if every member of a series of real numbers is less than x, then the limit of the series is less than x.
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Worng. My argument is that the long addition of the form 1/2 + 1/4 +1/8 ... is equivalent to long addition 0.12 + 0.012 + 0.0012 ... = 0.111...2 < 1 by 0.000...12, or in other words, no infinitely many added values are equal to the value of the given limit value 1, and this fact is very cool since we realize that the power of the non-composed segment 1 is greater than any amount of collection of added values along it (we are able to understand the non-entropic property of the association among the non-composed and the composed.

BenjaminTR said:
This is not true; if x is the limit this means that no matter how small a number epsilon you pick, you can always find a member of the series such that every later member of the series is less than epsilon away from x. From this definition it is provable that 1 is the limit as n approaches infinity of the series of sums 1/2 + 1/4 + ... 2-n. This is true even though every individual member of the series is less than 1. Consider: every finite integer is finite, but there are infinitely many of them. Infinity is very cool, and sets and series with infinitely many members have sometimes surprising properties.
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Again, the surprise is the non-entropic property of the association among the non-composed and the composed, that can't be known by a model that is based only on the composed (collections).
 
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doronshadmi said:
Worng. My argument is that the long addition of the form 1/2 + 1/4 +1/8 ... is equivalent to long addition 0.12 + 0.012 + 0.0012 ... = 0.111...2
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So far, so good.

... < 1 by 0.000...12
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And then it falls apart. You continually say it is less than one, but you never actually demonstrated it. Instead, you invent a meaningless notation for what the alleged but non-existence difference must be and believe you have accomplished something.

If 0.1111...2 were different from one, then you should be able to demonstrate that the difference is distinguishable from zero. Yet, you cannot. The best you have is gibberish like what I will now...

...<snip>...
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BenjaminTR said:
I am thinking visually and spatially. I am even looking at the diagram. It is plain to see that no matter how many circles you pile on, or how big or small they get, the circumference will always be greater than the diameter. Thus, we can conclude, using verbal-symbolic-visual-spatial reasoning that the length of the blue segment equals d only if the length of the purple curve is greater than d.
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Worng. the blue segment is less then d exactly because the added blue values along the blue segment are determined by (purple) semi-circles, which belongs to infinitely many levels of semi-circles, that are irreducible into d (each level of semi-circles has the invariant length d*(Pi/2)).

Moreover, the length of the purple semi-circles < d*(Pi/2), exactly because the most right semi-circle of any arbitrary given level of semi-circles is not added the length of the purple semi-circles, and this state holds upon infinitely many levels of semi-circles with length d*(Pi/2).

In other words, you still have to learn how the think visually and spatially.
 
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doronshadmi said:
Worng.
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Odd, because everything BenjaminTR posted was correct. What did you find wrong with it?

the blue segment is less then d exactly because the added blue values along the blue segment are determined by (purple) semi-circles, which belongs to infinitely many levels of semi-circles, that are irreducible into d (each level of semi-circles has the invariant length d*(Pi/2)).
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BenjaminTR sort of said that, just without all the gibberish, misused terms, and irrelevant asides.

Moreover, the length of the purple semi-circles < d*(Pi/2), exactly because the most right semi-circle of any arbitrary given level of semi-circles is not added the length of the purple semi-circles, and this state holds upon infinitely many levels of semi-circles with length d*(Pi/2).
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No, length of your violet path is less than its upper bound because it is finite part of an infinite sequence. It doesn't have all the parts, yet. The infinite sequence does reach the pi/2 value...and exactly the same is true for the blue length being equal to 1.

In other words, you still have to learn how the think visually and spatially.
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And yet, everything BenjaminTR posted was correct. Your posts, on the other hand, not so much.
 
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