Deeper than primes - Continuation

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doronshadmi said:
[...] where n = 1 → ∞ (where → means "approaches by infinitely many steps")).
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Doron, your symbolism is really garbage. You alter well-established symbolism and give it a different, counter-intuitive meaning, as if you wanted to obfuscate the works further still.
 
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epix said:
Lol. You are for an infinite wait, because Doron will not present his case without involving Doronetics.
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A: "There are things with red, green or blue colors, and if you mix them you get new colors"

B: "I may agree with you only if you show it by using black and white colors"
 
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What color is the last letter of "Doron"?

Hmm. Let's see... N is the 14th letter in the alphabet - like when you mix 1 and 4.
doronshadmi said:
A: "There are things with red, green or blue colors, and if you mix them you get new colors"

B: "I may agree with you only if you show it by using black and white colors"
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So if you mix black and white you get 1 color; when you mix red, green, and blue, you get 4 different colors. Then you mix it all together and you get 14 and that's the color of that last letter.

Okay, but what color is it then?

Let me see... wberuilgrbeteenladcke. I'd say the mix is a special shade of gibberish.

:p
 
doronshadmi said:
[...]



|{}|=0 > {} member (0 = something > nothing).



|{{}}|= 1 > |{}|=0 > {} member (0 = something > nothing).



|{{{}}}|= 2 > |{{}}|= 1 > |{}|=0 > {} member (0 = something > nothing).



etc... ad infinitum.



Now let's see the difference between actual infinity, potential infinity, finite and actual finite:



|N|= actual infinity = |{...{{{}}}...}| > |...{{{}}}...| = potential infinity = (... > |{{{}}}|= finite 2 > |{{}}|= finite 1 > |{}|= finite 0) > {} member (finite 0 > nothing that is actual finite).
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Is there any difference between your membership levels and the standard set theoretic heirarchy, with ur-elements? Just curious. They seem the same to me. You even have transfinite levels, like the standard one.



By the way, '0 > nothing' strikes me as nonsensical. The greater than relation is a relation between things like numbers. Unless you mean to deny the existence of negative numbers, this is semantically ill-formed. Do you just mean the number 0 exists? It would be easier to say it that way.



Moreover, we do not need sets in order to show the inaccessibility into a given limit.



For example the result of the long addition 1/2 + 1/4 + 1/8 + ... is inaccessible to limit 1.



So is the case of about the equivalent long addition 0.12 + 0.012 +0.0012 + ... = 0.111...2 ,which is inaccessible to limit 1.
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You claim to have shown that the sum of a series is not equal to its limit, but in fact you have merely stated it twice with no supporting argument.
 
BenjaminTR said:
Is there any difference between your membership levels and the standard set theoretic heirarchy, with ur-elements? Just curious. They seem the same to me. You even have transfinite levels, like the standard one.
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The level of set > the level of members, or their absence, for example:

The level the outer "{" and "}" is 0 and it is > than no level at all (nothing).

This reasoning is equivalent to the difference between nothing and a point, where a point is something at the level of 0-dimensianal space.

A Ur-element is something at level 0 > no level at all (nothing).

According to my reasoning the outer "{" and "}" is the set level, where this level is above members' membership.

As a result no amount of members is accessible to the set's level that, by definition, is above members' membership.

Given {...{{{}}}}...} expression, no amount of members' levels ...{{{}}}... is is accessible to the set's level (where the set's level is notated by the outer "{" and "}" of the expression {...{{{}}}}...}.

According to this reasoning, no set is its own member, and we are able to distinguish between the actual (the set's level) and the potential (the members level).

The actual level of a set is above the potential level of members, no matter if the amount of members' levels is finite or infinite.

In such a framework contradiction is impossible, simply because contradiction is derived only from things on the same level.

Moreover, infinitesimals are the distinct signature of the inaccessibility of the members' level into the set level (as seen in the case of {...{{{}}}...} expression).

BenjaminTR said:
By the way, '0 > nothing' strikes me as nonsensical. The greater than relation is a relation between things like numbers. Unless you mean to deny the existence of negative numbers, this is semantically ill-formed. Do you just mean the number 0 exists? It would be easier to say it that way.
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Negative numbers have nothing to do with cardinality (in the case of standard reasoning) or with levels (in the case of my non-standard reasoning), where is both cases < or > are used.

BenjaminTR said:
You claim to have shown that the sum of a series is not equal to its limit, but in fact you have merely stated it twice with no supporting argument.
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Since my non-standard reasoning of levels is equivalent the levels among dimensional spaces (where the term "dimensional spaces" is not restricted to Geometry or metric-spaces), an expression like {...{{{}}}...} asserts that ...{{{}}}... is the inaccessibility of any amount (amount of levels, in this case) to the outer "{" and "}" level of {...{{{}}}...} expression.

In terms of dimensional spaces, the outer "{" and "}" level of {...{{{}}}...} expression is irreducible into the ..{{{}}}... infinitely many levels, and the ..{{{}}}... infinitely many levels are non-extensible (inaccessible) to the outer "{" and "}" level.

An example of this non-standard reasoning (in terms of dimensional spaces, not restricted to Geometry or metric-spaces) is given in http://www.internationalskeptics.com/forums/showpost.php?p=9306456&postcount=2437.
 
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doronshadmi said:
The level of set > the level of members, or their absence, for example:

The level the outer "{" and "}" is 0 and it is > than no level at all (nothing).

This reasoning is equivalent to the difference between nothing and a point, where a point is something at the level of 0-dimensianal space.

A Ur-element is something at level 0 > no level at all (nothing).

According to my reasoning the outer "{" and "}" is the set level, where this level is above members' membership.

As a result no amount of members is accessible to the set's level that, by definition, is above members' membership.

Given {...{{{}}}}...} expression, no amount of members' levels ...{{{}}}... is is accessible to the set's level (where the set's level is notated by the outer "{" and "}" of the expression {...{{{}}}}...}.

According to this reasoning, no set is its own member, and we are able to distinguish between the actual (the set's level) and the potential (the members level).

The actual level of a set is above the potential level of members, no matter if the amount of members' levels is finite or infinite.

In such a framework contradiction is impossible, simply because contradiction is derived only from things on the same level.

Moreover, infinitesimals are the distinct signature of the inaccessibility of the members' level into the set level (as seen in the case of {...{{{}}}...} expression).


Negative numbers have nothing to do with cardinality (in the case of standard reasoning) or with levels (in the case of my non-standard reasoning), where is both cases < or > are used.



Since my non-standard reasoning of levels is equivalent the levels among dimensional spaces (where the term "dimensional spaces" is not restricted to Geometry or metric-spaces), an expression like {...{{{}}}...} asserts that ...{{{}}}... is the inaccessibility of any amount (amount of levels, in this case) to the outer "{" and "}" level of {...{{{}}}...} expression.

In terms of dimensional spaces, the outer "{" and "}" level of {...{{{}}}...} expression is irreducible into the ..{{{}}}... infinitely many levels, and the ..{{{}}}... infinitely many levels are non-extensible (inaccessible) to the outer "{" and "}" level.

An example of this non-standard reasoning (in terms of dimensional spaces, not restricted to Geometry or metric-spaces) is given in http://www.internationalskeptics.com/forums/showpost.php?p=9306456&postcount=2437.
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A brief essay on finding a common mathematical language by Palindromeus of Aachen.

level is to ---> and English as level is to <--- and Hebrew
 
doronshadmi said:
I prove the fallacy of the Limit, as used by standard analysis, which according to it, for example, the long addition 1/2 + 1/4 + 1/8 + ... = 1 (which is equivalent to the claim that 0.12 + 0.012 + 0.0012 + ... = 1)
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Doron, you think that proof and evidence are tight synonyms, but that's not so. You try to populate your argumentation with similarities, like invoking different number bases, hoping to gather as much quantitative evidence supporting your claim as possible, but that doesn't work that way and you should know that.

Zeno had noticed the same thing long before you made your own discovery. Nowadays, the argument gets powered by Peano induction axiom, if you can throw the line there. But that may prove treacherous, because no deviation from defined pattern can actually work against you, as you will see. The subject of Peano induction axiom relates to 0, 1, 2, 3, 4, ... and makes sure that there will be no other increment apart from 1 in the natural sequence. By implication, once a pattern is defined, it cannot be broken. Here is an example:

1 - 9/9 = 0.0000 ...
1 - 8/9 = 0.1111 ...
1 - 7/9 = 0.2222 ...
1 - 6/9 = 0.3333 ...
1 - 5/9 = 0.4444 ...
1 - 4/9 = 0.5555 ...
1 - 3/9 = 0.6666 ...

Now, when you see what the pattern is based on, you can safely continue.

1 - 2/9 = 0.7777 ...
1 - 1/9 = 0.8888 ...
1 - 0/9 = 0.9999 ...

All is good according to Peano. But the last term of the sequence offers an identity you wouldn't like to see around.

If 0/9 = 0, then 1 - 0/9 = 1 - 0.

Consequently, with respect to the last term of the pattern/sequence,

(1 - 0 = 0.9999 ...) => (1 = 0.9999 ...).

Then comes the proof that the definition of the sequence holds through and that 0.9999 ... = 1

x = 0.9999 ...
10x = 9.9999 ...
10x - x = 9.9999 ... - 0.9999 ...
9x = 9
x = 9/9 = 1

Unlike the language that makes doronetics, these arguments can be followed by a vast majority of folks and so you will have a hard time to sell your idea that the limit of the series in question cannot reach 1. You need to use similar language and arguments to counter.

Here is a path that may lead to your salvation. It starts here.
In Peano's original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section Models below.
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You need to switch from N0 to N1 and set up the product-unit axiom from which Peano induction axiom could be derived making it actually a theorem. Then you need to throw a little monkey wrench into the algebra proof.

(x = a) AND (x = 0.9999...)
x = a
10x = 10a
10x - x = 10a - a
9x = 9a
x = 9a / 9 = a
x = a = 0.9999 ...

Remember that the series converts itself into a sequence through cumulative addition and each of the term of the sequence is defined as

an = 1 - 1/2n for n = 1, 2, 3, 4, ...

The condition that includes the ellipses (...) is kept out of and so the ellipses symbolism cannot pollute the road leading toward the product-unit axiom.
 
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epix said:
Doron, you think that proof and evidence are tight synonyms, but that's not so. You try to populate your argumentation with similarities, like invoking different number bases, hoping to gather as much quantitative evidence supporting your claim as possible, but that doesn't work that way and you should know that.
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My argument is independent of any particular base, and you can know that if you stop your "jokes".

epix said:
The subject of Peano induction axiom ...
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Peano axioms have nothing to do with the inaccessibility of n-1 dimensional space to n dimensional space, or the irreducibility of n dimensional space into n-1 dimensional space (where the term dimensional space is not restricted to Geometry or Metric space).

epix said:
Unlike the language that makes doronetics, these arguments can be followed by a vast majority of folks and so you will have a hard time to sell your idea that the limit of the series in question cannot reach 1. You need to use similar language and arguments to counter.
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Unlike the language that makes Organic Mathematics (which is derived from verbal_symbolic AND visual_spatial reasoning) your arguments are derived from verbal_symbolic-only reasoning, and the current vast majority of folks are using verbal_symbolic-only reasoning, so they have a hard time to grasp the idea that the limit 1 is inaccessible to any amount of convergent long addition like 1/2 + 1/4 + 1/8 + ...

epix said:
You need to use similar language and arguments to counter.
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Wrong, you need to get out of your verbal_symbolic-only Limit(ed) box in order to get the mathematical universe of verbal_symbolic AND visual_spatial reasoning.

Once again you ignore http://www.internationalskeptics.com/forums/showpost.php?p=9306606&postcount=2443, and because of your "jokes" in http://www.internationalskeptics.com/forums/showpost.php?p=9306637&postcount=2445 you also have missed the analogy in http://www.internationalskeptics.com/forums/showpost.php?p=9306573&postcount=2442.

Look how poor is your reply (in http://www.internationalskeptics.com/forums/showpost.php?p=9310133&postcount=2449) to http://www.internationalskeptics.com/forums/showpost.php?p=9309433&postcount=2448.

epix said:
Zeno had noticed the same thing long before you made your own discovery.
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Zano had a partial understanding of the discussed subject, in order to realize this please look at http://www.scribd.com/doc/21967511/...considerations-of-Some-Mathematical-Paradigms.
 
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epix said:
The subject of Peano induction axiom relates to 0, 1, 2, 3, 4, ... and makes sure that there will be no other increment apart from 1 in the natural sequence. By implication, once a pattern is defined, it cannot be broken. Here is an example:

1 - 9/9 = 0.0000 ...
1 - 8/9 = 0.1111 ...
1 - 7/9 = 0.2222 ...
1 - 6/9 = 0.3333 ...
1 - 5/9 = 0.4444 ...
1 - 4/9 = 0.5555 ...
1 - 3/9 = 0.6666 ...

Now, when you see what the pattern is based on, you can safely continue.

1 - 2/9 = 0.7777 ...
1 - 1/9 = 0.8888 ...
1 - 0/9 = 0.9999 ...

All is good according to Peano. But the last term of the sequence offers an identity you wouldn't like to see around.

If 0/9 = 0, then 1 - 0/9 = 1 - 0.

Consequently, with respect to the last term of the pattern/sequence,

(1 - 0 = 0.9999 ...) => (1 = 0.9999 ...).
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Epix, take for example 1 - 3/9 :

1 - 3/9 = 1 - 1/3 = 2/3, where 2/3 is a local number and 0.666... is a non-local number < local number 2/3.

So is the case with all you wrote, and here is the correction, which is derived from the reasoning of inaccessible or irreducible dimensional spaces (where the term dimensional space is not restricted to Geometry or Metric space):

local number 1 - local number 9/9 = local number 0.0000 ...
local number 1 - local number 8/9 > non-local number 0.1111 ...
local number 1 - local number 7/9 > non-local number 0.2222 ...
local number 1 - local number 6/9 = local number 1 - local number 2/3 = local number 1/3 > non-local number 0.3333 ...
local number 1 - local number 5/9 > non-local number 0.4444 ...
local number 1 - local number 4/9 > non-local number 0.5555 ...
local number 1 - local number 3/9 = local number 1 - local number 1/3 = local number 2/3 > non-local number 0.6666 ...
local number 1 - local number 2/9 > non-local number 0.7777 ...
local number 1 - local number 1/9 > non-local number 0.8888 ...
local number 1 - local number 0/9 > non-local number 0.9999 ...

Moreover:

11.00100100001111...2 < 10.01021101222201...3 < 3.02100333122220...4 < 3.03232214303343...5 < 3.05033005141512...6 < 3.06636514320361...7 < 3.11037552421026...8 < 3.12418812407442...9 < 3.14159265358979...10 < 3.16150702865A48...11 < 3.184809493B9186...12 < 3.1AC1049052A2C7...13 < 3.1DA75CDA813752...14 < 3.21CD1DC46C2B7A...15 < 3.243F6A8885A300...16 ... < ... Pi, where Pi is a local number and the non-local numbers with infinitely many different bases (starting from base 2) < local number Pi.

In other words, you have no case in your verbal_symbolic-only box.


epix said:
x = 0.9999 ...
10x = 9.9999 ...
10x - x = 9.9999 ... - 0.9999 ...
9x = 9
x = 9/9 = 1
Click to expand...

Already discussed in

5961689020_3526c6e94d_o.jpg
 
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epix said:
The subject of Peano induction axiom relates to 0, 1, 2, 3, 4, ... and makes sure that there will be no other increment apart from 1 in the natural sequence. By implication, once a pattern is defined, it cannot be broken. Here is an example:

1 - 9/9 = 0.0000 ...
1 - 8/9 = 0.1111 ...
1 - 7/9 = 0.2222 ...
1 - 6/9 = 0.3333 ...
1 - 5/9 = 0.4444 ...
1 - 4/9 = 0.5555 ...
1 - 3/9 = 0.6666 ...

Now, when you see what the pattern is based on, you can safely continue.

1 - 2/9 = 0.7777 ...
1 - 1/9 = 0.8888 ...
1 - 0/9 = 0.9999 ...

All is good according to Peano. But the last term of the sequence offers an identity you wouldn't like to see around.

If 0/9 = 0, then 1 - 0/9 = 1 - 0.

Consequently, with respect to the last term of the pattern/sequence,

(1 - 0 = 0.9999 ...) => (1 = 0.9999 ...).
Click to expand...


Let us write it more clearly:

local number 1 - local number 9/9 = local number 0/9 = 0
local number 1 - local number 8/9 = local number 1/9 > non-local number 0.1111 ... by non-local number 0.000...910
local number 1 - local number 7/9 = local number 2/9 > non-local number 0.2222 ... by non-local number 0.000...810
local number 1 - local number 6/9 = local number 3/9 > non-local number 0.3333 ... by non-local number 0.000...710
local number 1 - local number 5/9 = local number 4/9 > non-local number 0.4444 ... by non-local number 0.000...610
local number 1 - local number 4/9 = local number 5/9 > non-local number 0.5555 ... by non-local number 0.000...510
local number 1 - local number 3/9 = local number 6/9 > non-local number 0.6666 ... by non-local number 0.000...410
local number 1 - local number 2/9 = local number 7/9 > non-local number 0.7777 ... by non-local number 0.000...310
local number 1 - local number 1/9 = local number 8/9 > non-local number 0.8888 ... by non-local number 0.000...210
local number 1 - local number 0/9 = local number 9/9 = 1 > non-local number 0.9999 ... by non-local number 0.000...110

So as you see, you have no case.
 
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Doron, you don't understand Zeno's Paradoxes. Why else are you having to rely upon quantum mechanics to explain it. In addition, with your most recent posts, you don't understand equality. Why do you keep falling back to local and non-local numbers? Why should 1 "local" have a value different of 1 "non-local"?
 
[Note: I do not know how to make the R for the real numbers here, so I just use 'R'.]

Doron, I am convinced your "Organic Mathematics" is based in part on a confusion about the connection between spaces and numbers. This manifests itself in the arguments you give against limits as the sums of series, or against 0.999... = 1. The arguments usually depend on your claim that no n-dimensional space is composed of n-1 dimensional spaces because in your system, spaces are not composed of parts at all, let alone of lower dimensional spaces or points. An argument where this comes up is the argument that 0.999... < 1 because a 1 dimensional space is not composed of points. You apparently find an extremely close relationship between spaces and numbers, otherwise this style of argument makes no sense at all. However, in claiming that spaces are not composed of points, you sever the connection required for these arguments to work.

Let's look at one reason why numbers and spaces tend to be closely connected in mathematics: many of the spaces people are interested in are composed of real numbers, or ordered tuples of real numbers. For example, where O is the set of all unions of open intervals of real numbers, (R, O) is a topological space. The space is made up of points, and the points are the real numbers themselves. Another example, if we let D(x, y) = |x-y|, for all real numbers x and y, then (R, D) is a metric space. Again, the space is composed of points, and the points are real numbers themselves. Similar things can be said of spaces based on Rn: these spaces are composed of points, which are ordered n-tuples of real numbers. Spaces based on real numbers are important because the nice properties of real numbers--like the fact that addition and multiplication are already defined, and that < is a linear order, etc--give the spaces based on them properties that are useful in all kinds of ways. The relation between these spaces and numbers arises from the fact that they are constructed out of numbers.

Further, even entities that are not explicitly composed of Rn often have important relations to Rn that are exploited. For example, manifolds in general relativity are not constructed of Rn, yet there must be bijections from regions of these manifolds to open subsets of Rn. In order for this to be possible, the manifold itself needs to be made of points that are matched to ordered tuples of real numbers by the bijection. Again we have a close connection between a kind of space and numbers that only exists because the spaces are made of points. Further examples are legion.

The bottom line is that if you reject the idea that spaces are composed, then none of these close connections between spaces and numbers exist. If spaces are not composed of real numbers or tuples of them, or at least is not composed of points with defined relations to Rn, then the fact that a space has a property does not show that numbers have a property. This is important. You have been making arguments of the form, "Spaces have property X, therefore numbers have property Y." But you have severed the normal connections between spaces and numbers. You have to do a great deal of work to show that there is a replacement connection between them. You have to show that there is a connection at all, then you have to show exactly what that connection is and why, and then you need to show that from this connection we can prove that we need to reconsider the properties and relations among numbers in such a way that, for example, 0.999... <> 1. So far, every one of these steps has been left out.

To make this a bit sharper, suppose you are right that spaces are not composed. Now we can show that we do not need to change anything about real analysis. First, you claim that points exist and line segments exist, along with all these other spaces. As far as I can tell, you allow these spaces to overlap, so there can be points on a line, though they are not part of the line; there can be line segments in a plane, though not part of it, etc. Thus, points overlap spaces. So far, there is not a number in sight. Numbers are not built in to spaces.

Now suppose we have an n-dimensional space S and want to define a distance metric over S. We need a function D from something related to S into the numbers (I leave open for now whether R is the relevant set of numbers or if we need some other set like the hyperreals or complex numbers). The problem is that S is a single thing, and it has no parts. Thus, we cannot use a function from parts of S to numbers, for there is only one part: the improper part, i.e. S itself (I guess we can define a trivial distance metric, that the distance between the space and itself is 0). Fortunately, there are points overlapping the space. These points, while not composing S, form a well defined set P, the set of all points overlapping S. We can now define a metric over space S in terms of P. We define a function from PxP to numbers, that satisfies the axioms of a metric, and that is our metric. Then distance between two objects that reside in S is defined in terms of which points in P overlap those objects. We started with non-composed spaces, but when trying to define distance we are back to a set of points.

Notice that this has no implications for the properties of the numbers. There is no reason yet to say that a function from PxP to Rn cannot be the required distance metric. All we have shown is that we are looking for a function from things that are not numbers to things that are. No such function is built into the space itself, so there are no implications whatsoever for the numbers.

Maybe you will now resort to claims about visual-spatial reasoning. "If we visualize a space and the points overlapping it," you might say, "we will see that Rn falls short; we can see that the points overlapping a space have a higher cardinality than the reals, so there is no function from the reals onto the points. This is because spaces are not composed, and points and never make up a higher dimensional space." However, this does not follow from visual-spatial reasoning, as far as I can tell. The only relevant visual/spatial "argument" I can imagine is to mentally "zoom in" to the level of individual points named by reals using some mapping and finding that there is space in between them. I do not know about you, but this stretches my visualization skills way past the breaking point. Between any two real numbers there are infinitely many more real numbers. I cannot even visualize that. I cannot determine through visualization whether we need something more than the reals to cover a space.

Even worse, it looks like your argument is: "The reals are not good enough because no matter which function from reals to points we choose, the range of the function does not compose the entire higher dimensional space." However, it is an axiom of your system that this is impossible. No set of points could ever possibly cover the whole space, by stipulation. Thus, this argument, if sound, would show that numbers of any kind are are ruled out by your system, since no matter what the domain, the range of points is axiomatically does not compose the whole space. I conclude instead that the argument is not valid.

Even if we could use some kind of superhuman visualization to see that there are more points than the reals can handle, this would not mean we would have to give up the reals. It would still be an option to ignore most of the points in P and define a metric for S only over a subset of P. Further, even if we decided we needed the hyperreals or some other system to do the job properly, the reals can still be defined in terms of these other structures. You just need to say, "The real numbers are defined as the set of numbers such that [axioms]." No problem. Then all of the conclusions of real analysis are true of this substructure, and nothing needs to be tossed out. At most, there would be an insanely minor revision in the understanding of what the reals are.

In conclusion, your arguments are invalid. In writing this, I have had to make several assumptions about "Organic Mathematics". They are the closest to coherent claims I could come up with that seemed to match doron's claims. Though largely pointless, writing this was surprisingly fun.
 
doronshadmi said:
Let us write it more clearly:

local number 1 - local number 9/9 = local number 0/9 = 0
local number 1 - local number 8/9 = local number 1/9 > non-local number 0.1111 ... by non-local number 0.000...910
local number 1 - local number 7/9 = local number 2/9 > non-local number 0.2222 ... by non-local number 0.000...810
local number 1 - local number 6/9 = local number 3/9 > non-local number 0.3333 ... by non-local number 0.000...710
local number 1 - local number 5/9 = local number 4/9 > non-local number 0.4444 ... by non-local number 0.000...610
local number 1 - local number 4/9 = local number 5/9 > non-local number 0.5555 ... by non-local number 0.000...510
local number 1 - local number 3/9 = local number 6/9 > non-local number 0.6666 ... by non-local number 0.000...410
local number 1 - local number 2/9 = local number 7/9 > non-local number 0.7777 ... by non-local number 0.000...310
local number 1 - local number 1/9 = local number 8/9 > non-local number 0.8888 ... by non-local number 0.000...210
local number 1 - local number 0/9 = local number 9/9 = 1 > non-local number 0.9999 ... by non-local number 0.000...110

So as you see, you have no case.
Click to expand...

The problem is that there is a relationship between p/q and a number expressed in the radix form. There is no apparent equivalency between, for example, 7/9 and 0.7777..., unless shown so. You explain it to a space alien this way: There is a 7 inches-long salami and you cut it on 9 equal parts. What is the length of one part?

If you have an x inches-long salami to equally divide, you concentrate on the word "long." That's because algorithm called "the long division" is the knife you will use. The process of measuring is sequential and the result looks this way.

7/9 --> LD(7, 9) = 0.7777...

You shouldn't get overly pedantic with the precision, otherwise you would spend an infinite amount of time to come up with a precise measurement.

Now it comes down to this: If the expression 7/9 holds equivalency with the long division, meaning that 7/9 is just another way to write LD(7, 9), then

7/9 = 0.7777...

and you don't have a case with your 7/9 > 0.7777..., especially when the statement is supported by nothing but categorical hierarchy of local and non-local numbers, which is something that doesn't exist outside doronetics.

Mathematicians discovered a long time ago that when an expression is followed by ellipses, nothing ever gets done, because, like in the case of 0.7777..., the point that represents the value on the R line is constantly moving (it is non-local; it travels.) And so, since 0.7777... is the result of LD(7, 9), they got rid of the ellipses by the association between LD(7, 9) and the expression 7/9.

There is no way of comparing 7/9 with 0.7777... unless both expressions are in the same format - be it a ratio or the radix form.

So you are saying that 1(local) > 0.9999...(non-local). Since 0.9999... is a periodic number, it is a rational number and as such, there must exist p/q = 0.9999... Find that p/q so we can compare it with 1/1 to see if you are right or not.
 
BenjaminTR said:
You apparently find an extremely close relationship between spaces and numbers, otherwise this style of argument makes no sense at all. However, in claiming that spaces are not composed of points, you sever the connection required for these arguments to work.
Click to expand...

It works as follows: a non-composed space > 0 is measured as actual infinity w.r.t any amount of lower spaces or sub-spaces, which are entirely at its domain, but they are no more than potential infinity w.r.t it.

As a result any given R member, is a local number that is surrounded by infinitely many levels of infinitely many infinitesimals at each level (which are not R members), where infinitesimal numbers are the distinct signature of the inaccessibility of collections of lower spaces to the level of actual infinity of a given higher non-composed space.

In your mathematical universe there are no infinitesimals and the are no higher non-composed spaces that are inaccessible to any amount of lower spaces, so you can't use your mathematical universe in order to conclude anything about Organic Mathematics universe.

As long as your reasoning gets actual infinity at the level of collections, you simply have nothing to say about Organic Mathematics.

BenjaminTR said:
Between any two real numbers there are infinitely many more real numbers. I cannot even visualize that. I cannot determine through visualization whether we need something more than the reals to cover a space.
Click to expand...
If there are infinitely many more real numbers, between any two real numbers, and a real number is equivalent to a given point, then one easily concludes that the term more is invariant upon infinitely many scale levels along a given non-composed line.

In other words, no amount of reals, hyperreals, hyperhyperreals, hyperhpyperhpyperreals, ... etc. ad infinitum are able to completely cover a non-composed given line.

As for practical use, one can always filter out objects of collections in order to get a requested result, but it does not not change the argument that collections are no more than potential infinity w.r.t higher non-composed space, and a given non-composed space > 0 is not less than actual infinity w.r.t collection of lower spaces or sub-spaces, which enables one to essentially understand non-entropic realm, which is essentially important to for further development of complex creatures (and we are some case of complex creatures).

BenjaminTR said:
I cannot even visualize that. I cannot determine through visualization whether we need something more than the reals to cover a space.
Click to expand...
As long as your "visualization" is actually verbal_symbolic-only reasoning.

BenjaminTR said:
writing this was surprisingly fun.
Click to expand...
For me it is not fun to see an intelligent person that is closed under verbal_symbolic-only reasoning.
 
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epix said:
So you are saying that 1(local) > 0.9999...(non-local). Since 0.9999... is a periodic number, it is a rational number and as such, there must exist p/q = 0.9999... Find that p/q so we can compare it with 1/1 to see if you are right or not.
Click to expand...

# = placeholder for some digit.

0.###... is not a rational number, where p/q is a rational number.

epix said:
There problem is that there is a relationship between p/q and a number expressed in the radix form.
Click to expand...
There is no problem, p/q > 0.###... , where 0.###... is simply the the inaccessibility of collections of lower spaces or sub-space to reach the level of actual infinity of a given higher non-composed space.
 
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Little 10 Toes said:
Doron, you don't understand Zeno's Paradoxes.
Click to expand...
Show it in details according to http://www.scribd.com/doc/21967511/...considerations-of-Some-Mathematical-Paradigms.

Little 10 Toes said:
Why else are you having to rely upon quantum mechanics to explain it.
Click to expand...
SRT, not QM.

Little 10 Toes said:
In addition, with your most recent posts, you don't understand equality. Why do you keep falling back to local and non-local numbers? Why should 1 "local" have a value different of 1 "non-local"?
Click to expand...
You do not understand the inequality between R members (which are all local numbers) and non-local numbers (which are not R members).
 
Here is how verbal_symbolic AND visual_spatial reasoning really works:

First, let's use again this quote:

doronshadmi said:
In the following diagram

9076101160_b50490da84_o.jpg


the length of the semi-circles is the constant value d*(Pi/2) upon infinite amount of levels, exactly because a 2-dim space (the area of each semi-circle in this diagram) is irreducible into 1-dim (the limit line with length d) (there is no homomorphism between n dimensional space and n-1 dimensional space, where n = 1 → ∞ (where → means "approaches by infinitely many steps")).

So is the case about the size of the added areas under infinitely many levels of semi-circles (starting form the second level), where this size is calculated by the long addition (1/2 of the area under the first semi-circle) + (1/4 of the area under the first semi-circle) + (1/8 of the area under the first semi-circle) + (1/16 of the area under the first semi-circle) + (1/32 of the area under the first semi-circle) + ... < (1/1 area under the first semi-circle).

As long at the notion of levels is not used by a given reasoning, this reasoning can't avoid contradiction.

Standard Analysis and ZF(C) are two examples of single-level reasoning.
Click to expand...

By forcing Standard Analysis, the potential infinity upon infinitely many levels (with the invariant length d*(Pi/2) of the semi-circles, and the convergent areas of infinitely many levels of semi-circles (as observed in the quote)) is collapsed into a single-level of 1-dimesional space with length d by quantum leap, and we are left with a finite collection of levels of semi-circles with length d*(Pi/2), a finite collection of levels of semi-circles' areas, and a single limit line with length d.
 
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doronshadmi said:
# = placeholder for some digit.

0.###... is not a rational number, where p/q is a rational number.
Click to expand...


Just because # is a placeholder for a single digit doesn't mean that all numbers displayed on a decimal scale that trail ellipses are not rational. If you were familiar with algorithm called "continued fractions," you would immediately notice that the results branch into two main categories,

1) the process is finite
2) the process is infinite

where (1) involves only rational numbers and (2) only irrational numbers. When you investigate the cause of the distinct separation, you find out that it is the periodicity of rational numbers that causes the algorithm to stop (on zero: stop). If you still insist on regarding all non-local numbers as irrational, then we can expect the clash of the titans.
In arithmetic, repeating decimal is a way of representing a rational number. Thus, a decimal representation of a number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely. For example, the decimal representation of 1/3 = 0.3333333… or 0.3 (spoken as "0.3 repeating", or "0.3 recurring") becomes periodic just after the decimal point, repeating the single-digit sequence "3" infinitely. A somewhat more complicated example is 3227/555 = 5.8144144144…, where the decimal representation becomes periodic at the second digit after the decimal point, repeating the sequence of digits "144" indefinitely.

Rational numbers are numbers that can be expressed in the form a/b where a and b are integers and b is non-zero. This form is known as a common fraction. On the one hand, the decimal representation of a rational number is ultimately periodic, as explained below. On the other hand every real number which has an eventually periodic decimal expansion is a rational number. In other words the numbers with eventually repeating decimal expansions are exactly the rational numbers (i.e.: those that can be expressed as ratios).
Click to expand...

The bold text doesn't mention anything about a "proven way," but unlike in doronetics, which doesn't know how to convert a ratio into a decimal number, standard math does.

So we have learned so far that doronetics labels 0.3333... as an irrational number despite the fact that the decimal number has a strong relation with ratio 1/3. The reason for this surprising conclusion is, as I already mentioned it, that doronetics doesn't understand the process called the long division. (The adjective "long" is derived from the fact that it sometimes take an infinite amount of time to get to the "final" result, like in the case of 1/3, for example.)

A decimal that is neither terminating [local] nor repeating [non-local] represents an irrational number (which cannot be expressed as a fraction of two integers), such as the square root of 2 or the number π. Conversely, an irrational number always has a non-repeating decimal representation.
Click to expand...
There is no problem, p/q > 0.###... , where 0.###... is simply the the inaccessibility of collections of lower spaces or sub-space to reach the level of actual infinity of a given higher non-composed space.
Click to expand...
The inequality p/q > 0.###... happens to be good for any positive p/q.

In conclusion, doronetics treats periodic decimals as irrational numbers.
 
epix said:
In conclusion, doronetics treats periodic decimals as irrational numbers.
Click to expand...
Wrong conclusion.

According Organic Mathematics there are local numbers or non-local numbers.

The non-local numbers can be periodic or non-periodic, and no one of these numbers is a member of R set (which includes only local numbers), and as you know, R set includes also the members of Q and N sets.
 
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doronshadmi said:
Wrong conclusion.

According Organic Mathematics there are local numbers or non-local numbers.

The non-local numbers can be periodic or non-periodic, and no one of these numbers is a member of R set (which includes only local numbers), and as you know, R set includes also the members of Q and N sets.
Click to expand...

Please define your R set, as it's clearly not the set of reals. And by "define" I don't mean "it's the set of locals". You say that it includes Q and N. Assuming that Q is the set of rational numbers, then R includes 0.(3). Is that local? :boggled:

Also, please define some operations on the set so we can put it to some use. You know, like for example addition and multiplication. Is the sum of two locals also a local? How about the product?
 
In addition to http://www.internationalskeptics.com/forums/showpost.php?p=9315683&postcount=2461, line d is included but no amount of area's levels is reducible into that line.

So we actually have two options:

1) The amount of levels is finite and the limit line with length d is included.

2) The amount of levels is infinite and the limit line with length d is included.

In both cases the areas are irreducible into the limit line with length d.

In terms of Zeno's Achilles\Tortoise Race, there is no race if:

1) There is a path but Achilles and Tortoise are not along it.

2) There is a point (or a limit point) but then Achilles and Tortoise do not exist as two comparators (this is the realm of Standard Analysis, but since there is no race in the first place, Standard Analysis has no case).

So the race is possible only if the realm is a mixing among the non-locality of a non-composed line and the locality of a non-composed point.

At this mixing realm, Achilles wins by finitely many steps or Achilles does not win along infinitely many convergent steps (as can be seen in http://www.scribd.com/doc/21967511/...considerations-of-Some-Mathematical-Paradigms).
 
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laca said:
Please define your R set,
Click to expand...
R is the standard set of real numbers, and no one of these numbers is a non-local number.

One knows non-local numbers only if he/she understands the irreducibility of a higher dimensional space into lower dimensional spaces, or the non-extensibility of lower dimensional spaces into a higher dimensional space.

As for multiplication of non-local numbers, for example 0.333...10*2 = 0.666...10, where 0.666...10 < 2/3 by 0.000...410, as clearly given in http://www.internationalskeptics.com/forums/showpost.php?p=9313916&postcount=2454.
 
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doronshadmi said:
Let us write it more clearly:

local number 1 - local number 9/9 = local number 0/9 = 0
local number 1 - local number 8/9 = local number 1/9 > non-local number 0.1111 ... by non-local number 0.000...910
local number 1 - local number 7/9 = local number 2/9 > non-local number 0.2222 ... by non-local number 0.000...810
local number 1 - local number 6/9 = local number 3/9 > non-local number 0.3333 ... by non-local number 0.000...710
local number 1 - local number 5/9 = local number 4/9 > non-local number 0.4444 ... by non-local number 0.000...610
local number 1 - local number 4/9 = local number 5/9 > non-local number 0.5555 ... by non-local number 0.000...510
local number 1 - local number 3/9 = local number 6/9 > non-local number 0.6666 ... by non-local number 0.000...410
local number 1 - local number 2/9 = local number 7/9 > non-local number 0.7777 ... by non-local number 0.000...310
local number 1 - local number 1/9 = local number 8/9 > non-local number 0.8888 ... by non-local number 0.000...210
local number 1 - local number 0/9 = local number 9/9 = 1 > non-local number 0.9999 ... by non-local number 0.000...110
So as you see, you have no case.
Click to expand...

I think you've just proved that your "non-local number" 0.000...110 is equal to zero.
 
doronshadmi said:
[...]

In your mathematical universe there are no infinitesimals and the are no higher non-composed spaces that are inaccessible to any amount of lower spaces, so you can't use your mathematical universe in order to conclude anything about Organic Mathematics universe.

[...]
Click to expand...

I don't have a problem with infinitesimals per se, or with non-composed spaces per se. There already exist rigorous treatments of infinitesimals, so it would be foolish to reject them entirely. In fact, I believe you can construct a set obeying the axioms of the hyperreals using set theory. While I have never heard of a rigorous treatment of non-composed spaces, at the moment I suspect it is possible to produce one. In fact, my previous post included an attempt to take a baby step in that direction.



The problem I have is with your attempted proofs based on these ideas. They seem confused, for reasons I, jsfisher and others have enumerated. In short, assuming your basic premises are true, your proofs still do not work. They are invalid.



In other words, no amount of reals, hyperreals, hyperhyperreals, hyperhpyperhpyperreals, ... etc. ad infinitum are able to completely cover a non-composed given line.



As for practical use, one can always filter out objects of collections in order to get a requested result, but it does not not change the argument that collections are no more than potential infinity w.r.t higher non-composed space, and a given non-composed space > 0 is not less than actual infinity w.r.t collection of lower spaces or sub-spaces, which enables one to essentially understand non-entropic realm, which is essentially important to for further development of complex creatures (and we are some case of complex creatures).



[...].
Click to expand...

Much if this is exactly what I said. If spaces are not composed, then you can never put together enough points to compose an entire space (unless the space is a point). This is an axiom of your system, if I understand correctly. The question is how this has anything to do with numbers.



You need to define some kind of mapping between spaces and numbers before there is any relation between spaces and numbers. The nature of the numbers is independent from the nature of the non-composed spaces you are talking about. Conclusions about spaces alone cannot imply anything about the numbers themselves; they can only imply restrictions on the types of relations that can exist between numbers and spaces.



The arguments about 0.999... are arguments about numbers and numbering systems, not about spaces. Conclusions about spaces cannot settle this. You need a proof that numbers themselves must have some property or relation, and that they can only have that if 0.999... < 1. This is exactly what has been lacking.
 
doronshadmi said:
R is the standard set of real numbers, and no one of these numbers is a non-local number.

One knows non-local numbers only if he/she understands the irreducibility of a higher dimensional space into lower dimensional spaces, or the non-extensibility of lower dimensional spaces into a higher dimensional space.

As for multiplication of non-local numbers, for example 0.333...10*2 = 0.666...10, where 0.666...10 < 2/3 by 0.000...410, as clearly given in http://www.internationalskeptics.com/forums/showpost.php?p=9313916&postcount=2454.
Click to expand...
In other words, you are saying that

2/3 - 0.666... = 0.000...4

I already told you that numbers can be compared only when they appear in the same numerical format. You have one number (0.666...) in the decimal format and the other (2/3) in the ratio, or the fractional format. That's like comparing temperatures listed in different scales. Just select one of the formats and do the necessary conversion. Do it step by step and don't involve any propositions concerning spaces nor include any link where the requested has been already "clearly" explained.

The decimal format came to existence with the birth of irrational numbers, which are numbers that cannot be expressed by ratio p/q. So when you want to compare values of two numbers where one is a rational number and the other is an irrational number, then you have to convert the rational number p/q into its decimal equivalent. Since the numbers in your inequality are all rational (by the standard math view), convert 0.666... and 0.000...4 into the fractional form.

It would be easier to list all terms in the decimal format due to the majority of the terms in the whole expression, but that would prove your inequality false, because

2/3 => LD(2, 3) = 0.666... (where LD means Long Division algorithm.)
http://en.wikipedia.org/wiki/Long_division

So why don't you demonstrate the way Organic Mathematics converts 2/3 into its decimal equivalent?
 
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epix said:
In other words, you are saying that





2/3 - 0.666... = 0.000...4





I already told you that numbers can be compared only when they appear in the same numerical format. You have one number (0.666...) in the decimal format and the other (2/3) in the ratio, or the fractional format. That's like comparing temperatures listed in different scales.
Click to expand...


This is not quite true. For example, given that 1/4 < 0.3 and 0.3 < log102, we can conclude that 1/4 < log102 without using the same format. These are just names for numbers, after all; they are not parts of the numbers themselves. Doron's methods are suspect, but for other reasons.
 
BenjaminTR said:
This is not quite true. For example, given that 1/4 < 0.3 and 0.3 < log102, we can conclude that 1/4 < log102 without using the same format. These are just names for numbers, after all; they are not parts of the numbers themselves. Doron's methods are suspect, but for other reasons.
Click to expand...


And there you hit at the heart and depth of Doron's nonsense. For him, "2" isn't a symbol representation of the number 2, an abstract concept; it is the number itself. 1/4 is also a number, and it must therefore be different from the different in appearance 0.25. Doron, himself, has said exactly that.

Doron completely avoids dealing with 0.999... as a representation of a number as a power series. It is necessary to preserve the fantasy 0.999... differs from 1. Moreover, since for Doron, notation is the number, gibberish like 0.000...1, although devoid of Mathematical meaning, gives Doron great comfort and a false believe he has discovered something profound.
 
BenjaminTR said:
This is not quite true. For example, given that 1/4 < 0.3 and 0.3 < log102, we can conclude that 1/4 < log102 without using the same format. These are just names for numbers, after all; they are not parts of the numbers themselves. Doron's methods are suspect, but for other reasons.
Click to expand...

Not only that you made the same mistake as Doron, but you made it even worse. In order to show that the inequality 0.3 < log10(2) is true, you need to convert the expression log102 in the exact format into the approximate (decimal) format. Only after

log10(2) = 0.30102999566...

you do the comparison, and indeed

0.3 < 0.30102999566...

"Given that 1/4 < 0.3" is good only after you do the subtraction which houses the proof. That means you need to use a common numerical format. For example, in the exact format

1/4 - 3/10 = -1/20

and that proves 1/4 < 0.3.
 
epix said:
Not only that you made the same mistake as Doron, but you made it even worse. In order to show that the inequality 0.3 < log10(2) is true, you need to convert the expression log102 in the exact format into the approximate (decimal) format....
Click to expand...

I think you misread what BenjaminTR wrote.
 
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