Cont: Deeper than primes - Continuation 2

Status
Not open for further replies.
Wholeness is not necessarily Comleteness

Wholeness is not necessarily Completeness, as seen in www.internationalskeptics.com/forums/showpost.php?p=12043625&postcount=2798 exactly because infinitely many things are infinitely weaker that actual infinity (as seen in www.internationalskeptics.com/forums/showpost.php?p=12412827&postcount=3095).

In order to deal with such notions, philosophy and mathematics are inseparable of each other (http://www.internationalskeptics.com/forums/showpost.php?p=12569431&postcount=3280).
 
Last edited:
Please look at the following diagram:

32803839331_176b1198a5_o.jpg


It was known as "2X=X√2 paradox" (This is an old "problem" that was known at least to Leibniz and probably to the Greeks).

Actually, this is not a paradox at all since no integer is an irrational number, and a straightforward way to show it, is by X=1, that is, 2>√2.

By observing the top of the attached diagram, one finds the convergent series a+b+c+d+...

1) Please pay attention that this series is rigorously defined by the intersections of the black straight lines (which go through the peaks of the zig-zag (black, red, green, magenta, blue, cyan) lines with constant length 2X) with each side of the square.

2) It means that the mathematical fact that 2X>X√2, is inseparable of the mathematical fact that 2X>2(a+b+c+d+...).

Let X (one side of the square) = 1

In that case (a+b+c+d+...) is actually (1/2+1/4+1/8+1/16...).

By (2) 2(1)>2(1/2+1/4+1/8+1/16...), which can be reduced into 1>1/2+1/4+1/8+1/16...

It has to be stressed that no partial sums like a, a+b, a+b+c, ... are involved in this argument, but not less than the series a+b+c+d+...

If one does not agree with the argument above, one has to prove (according to the considered diagram) that series a+b+c+d+... is not defined by the zig-zag lines (where, again, no partial sums like a, a+b, a+b+c, ... are involved in such proof).

Moreover, if one proves it, one also demonstrates why visualization is insufficient for rigorous mathematical results.

I am fully aware that what is called "not a summation in the usual sense" means a+b+c+d+... ≤ X, where the semantics (meaning) of ≤ (in the considered case) is "not greater than" X, or "at most" X. Since series a+b+c+d+... is strictly defined by all the zig-zag lines such that 2X is strictly > X√2, series a+b+c+d+... can't be but strictly < X. So I still do not see how ≤ is relevant to the diagram above.
 
Last edited:
I wish to stress that, for example:

S = 1/2+1/4+1/8+1/16...

2S = 1+1/2+1/4+1/8+1/16...

2S - S = 1 - S

is not a proof of the considered case because:

1) By omitting S from 2S there is no guarantee that the omitted value (= 1/2+1/4+1/8+1/16...) is equal to the non-omitted value (= 1).

2) The separability between 2>√2 and 1>1/2+1/4+1/8+1/16... has not been proven.
 
Last edited:
The standard notion of set (according to Prof. Melvin Randall Holmes):
A set is a collection determined by its elements. Finite sets are often written {a, b, c} (for example), by listing their elements. Order does not matter and repeated items do not change the intended meaning.

The elements of the sets are not parts of the set. The set is not made by conglomerating its elements together. This is a common misunderstanding.

To see this it is enough to play with the notation. {x} is not the same object as x: if a set were made up of its elements as parts, this would not make sense. If you don’t believe this, look at {{2, 3}}: this is a set with one element, while its sole element is a set with two elements, so they are different.

Another way of seeing it is to notice that a relation of part to whole should be transitive. If a is part of b and b is part of c, then a is part of c. But notice that 2 ∈ {2, 3} and {2, 3} ∈ {{2, 3}}, but 2 is not a member of {{2, 3}}
Click to expand...



By logically going beyond the notion of collection |{}| is tautology and {||} is contradiction, such that any given collection is ~contradiction AND ~tautology.

As about cardinality:

{||} = 0

|{}| = ∞ = the cardinality of actual infinity

{|...|} = any cardinality > 0 AND < ∞


Some examples:

{|{}|} = 1

|{{}}| = ∞

{{||}} = 0

{|{1,2}|} = 1

{{|1,2|}} = 2

{|1,2|} = 2

{{1,{||},2}} = 0

|{{1,{},2}}| = ∞


Nested cardinality examples:

|{|{|1,{||},2|}|}| = (((0)3)1)∞

|{|{|1|,{||},2}|}| = (((0)1)1)∞

|{|{|1|,{||},|2|}|}| = (((0)1,1)1)∞

|{|{|1|,|{||}|,|2|}|}| = (((0)1,1,1)1)∞

|{||}| = (0)∞

etc. ...

-------------------

As can be seen, the standard notion of collection is a very limited mathematical framework.
 
Last edited:
By going beyond the notion of collection (which is a composed thing) the non-composed is defined by non-composed opposite extremes, which are NOthing and YESthing, where the cardinalities (the magnitudes) of them are |{||}| = (0)∞

So the cardinality of any give collection is > 0 AND < ∞, which means that no collection is accessible to that has cardinality 0 (NOthing) or cardinality ∞ (YESthing).

By being aware of the composed and the non-composed, one enables to understand why a collection with endless members is not actual infinity, simply because it is inaccessible to YESthing (that has cardinality ∞).
 
Last edited:
Apathia said:
Yes. To put it in other words: Actual Infinity is Being. It's not a subject, object, or existing thing. Your Thing is Being, Awareness without an object.
As I said, the words get in the way. If we talk about Being we make an object of it and immediately lose it to existence. You can only Be it.

I want to communicate to others reading this what UNITY awareness feels like. My best example is interpersonal, and I'm afraid that a number of readers still won't see the direction its pointing. But there are times, sometimes with meditation and others just suddenly, that I feel in Love, but my Love has no object or objects. There's no person or group who is the Loved. Yet Love seamlessly reaches all. No one is singled out as the object of that Love, and yet there is no one who is not in that Love.

But this is a poor way of putting it, because you have to understand the Love is not some kind of light that shines from me. There's no subject either. There is the shining of Being. I just Love. I'm just Loving.

For the individuals I meet, they feel Love from me. They don't feel they are an object of some romantic agenda or confidence game. (At least many don't. I've met people who are suspiciously sure that that charisma is a ploy. I've also met those whose hearts resonated to that seamless compassion and were frightened that the UNITY was dissolving their ego identities. Such defensive people then accuse me of being "needy," or having poor personal boundaries. I give them their desired distance.)

I Love the person before me, not as the object of love, but as a participant in Being and UNITY.

If we speak of others, though we are all of each other, it seems the light does not shine from me but from the other. Whoever is before me at that moment is the front and center of all.

I would like to see everyone cultivating this UNITY awareness. It's the only real anecdote to our contemporary society where individuals are objects and commodities of marketing and xenophobic fear of others and outsiders has become the chief political motivation.
Click to expand...

Apathia this is a beautiful essential sharing.
 
Doron, you are clearly afraid of the implications of palindromic number summation. You do not surprise me in this.

But if you can overcome your timidity and resume silence, much can still be achieved.
 
doronshadmi,

If you want to have a discussion with someone here, make sure that that someone is here to participate in the discussion. Comments made elsewhere, such as in response to a Youtube video, should be discussed at that elsewhere. You may not import statements from that elsewhere in the guise of continuing a discussion here, especially when it is clear your intent is simply to reintroduce an argument from the past.

Thirty-four posts have been shuffled off to AAH.
Replying to this modbox in thread will be off topic  Posted By: jsfisher
 
Last edited:
Please very carefully observe Prof. Edward Frenkel's YT video on vectors https://www.youtube.com/watch?v=PFkZGpN4wmM .

The essence of his notion is given from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video, where he introduces the idea which I call "Math Over Matrix", which is actually our ability to deduce also beyond the notion of collections (the mug is not a collection of its projections).

I find that his idea of the difference between "the thing in itself" and its representations (he is an expert in Representation Theory) mathematically airs its view by the following two axioms:

(1) The Axiom Of Non-Complexity: There exists, at least, object _ (1-dim object), such that it is not a collection of shorter or shortest objects. (a shortest object is at least . (0-dim object)).

(2) The Axiom Of Markers: Given a collection of shorter or shortest objects, they define values with respect to, at least, object _

Without (1) there is no, at least, object _ , and without (2) no value can be defined with respect to, at least, object _

So a useful mathematical framework, in this case, is based on, at least, (1) AND (2).

Please pay attention that (1) and (2) are deduced by using both visual_spatial AND verbal_symbolic reasoning, which actually enables to distinguish between "the thing in itself" (the non-composed existence, given by visual_spatial) and a collection (the composed, given by verbal_symbolic reasoning) of representations that are related to it but they are not the same as the "the thing in itself" (the mug ("the thing in itself") is not a collection of its projections (it is not its representations)).

Please pay attention that _ or . is non-composed, where this common property actually enables _ . deduction by the same framework.

But in order to do useful deduction, _ or . are also distinguishable of each other such that at least _ (1-dim object) is irreducible into at least . (0-dim object) AND at least . (0-dim object) is not expansible into at least _ (1-dim object).

Now let's deduce spatially AND symbolically according to (1) and (2) by using what is known as positional numeral system (if only verbal_symbolic reasoning is used).

Since the considered mathematical framework is based on at least visual_spatial AND verbal_symbolic reasoning, one actually uses positional number system.

Let's observe, for example 0.111...[base 2] and 0.222...[base 3] (which are numbers of their own that are not equal to 1, if visual_spatial AND verbal_symbolic reasoning is used), by the following visual_spatial AND verbal_symbolic diagram:

32201137757_dd547db90d_o.jpg


Since at least _ (1-dim object) is irreducible into at least . (0-dim object), for example, 0.222...[base 3] < 1 by at least 0.000...1[base 3], then the ...1 in 0.000...1[base 3] is exactly the irreducibly of at least _ (1-dim object) into at least . (0-dim object).

By using visual_spatial AND verbal_symbolic reasoning, the notion of invariant proportion is used among potential infinitely many scales (where potential infinity is simply the result of at least _ (1-dim object) that is irreducible into at least . (0-dim object)) as follows:

0.222...[base 3] < 1 by at least 0.000...1[base 3], but this is only the least case under [base 3].

Now, please (by using visual_spatial AND verbal_symbolic reasoning) very carefully pay attention to the potential infinitely many stairs in 0.222...[base 3] along the diagram above.

By doing so we get the following:

0.222...[base 3] < 1 by at least 0.000...1[base 3]

but

0.22...[base 3] < 1 by at least 0.000...2[base 3]

where

the invariant proportion among 0.000...2[base 3] (___) and 0.000...1[base 3] (_) is kept along the diagram, exactly because _ or ___ (1-dim form) are irreducible into . (0-dim form).

___ is 3 times longer than _ by the invariant proportion along the potential infinitely many scales of the [base 3] positional number system (of the example above).

[base 3] positional number system is used here for any [base > 1] positional number system, without loss of generality ( https://en.wikipedia.org/wiki/Without_loss_of_generality ).
 
Last edited:
In the previous post I wrote:

doronshadmi said:
By doing so we get the following:

0.222...[base 3] < 1 by at least 0.000...1[base 3]

but

0.22...[base 3] < 1 by at least 0.000...2[base 3]

where

the invariant proportion among 0.000...2[base 3] (___) and 0.000...1[base 3] (_) is kept along the diagram, exactly because _ or ___ (1-dim forms) are irreducible into . (0-dim form).

___ is 3 times longer than _ by the invariant proportion along the potential infinitely many scales of the [base 3] positional number system (of the example above).
Click to expand...

By more careful observation, I have realized that ...x[base y] form has to be involved with the power value of some base value along the potential infinitely many scales of a given positional number system (without being confused with the basepower at the left side of the radix point), so the right representation of ...n[base y] form (as done above) has to be as follows:

Let b be the base value, which is represented by any natural number > 1

Let p be the power value of b, which is represented by any natural number ≥ 0

The general form is ...bp
If (for example) b = 3, then a better representation of an invariant proportion along potential infinitely many scales (of this case) is done as follows:

0.000...1[base 3] is actually ...30
0.000...2[base 3] is actually ...31
etc.

(I knew that I used 3 as a part of [base 3], but it is done as a "short cut" in order to air my view more simply at first glance)

Now let's rewrite the quote by using the new symbolic representation (including some highlighted corrections):

doronshadmi said:
By doing so we get the following:

If 0.222...3 < 1 by at least 0.000...30
then
0.22...3 < 1 by at least 0.000...31
where

the invariant proportion among 0.000...31 (___) and 0.000...30 (_) is kept along the diagram, exactly because _ or ___ (1-dim forms) are irreducible into . (0-dim form).

___ is 3 times longer than _ by the invariant proportion along the potential infinitely many scales of the [base 3] positional number system (of the example above).
Click to expand...
 
Last edited:
I wish to correct the following part:

doronshadmi said:
I find that his idea of the difference between "the thing in itself" and its representations (he is an expert in Representation Theory) mathematically airs its view by the following two axioms:

(1) The Axiom Of Non-Complexity: There exists, at least, object _ (1-dim object), such that it is not a collection of shorter or shortest objects. (a shortest object is at least . (0-dim object)).

(2) The Axiom Of Markers: Given a collection of shorter or shortest objects, they define values with respect to, at least, object _

Without (1) there is no, at least, object _ , and without (2) no value can be defined with respect to, at least, object _

So a useful mathematical framework, in this case, is based on, at least, (1) AND (2).
Click to expand...


In order to not be closed under object|subject duality it has to be:

doronshadmi said:
I find that his idea of the difference between "the thing in itself" and its representations (he is an expert in Representation Theory) mathematically airs its view by the following two axioms:

(1) The Axiom Of Non-Complexity: There exists, at least, _ (1-dim thing), such that it is not a collection of shorter or shortest things. (a shortest thing is at least . (0-dim thing)).

(2) The Axiom Of Markers: Given a collection of shorter or shortest things, they define values with respect to, at least _ thing.

Without (1) there is no, at least, _ thing , and without (2) no value can be defined with respect to, at least, _ thing.

So a useful mathematical framework, in this case, is based on, at least, (1) AND (2).
Click to expand...
 
Last edited:
Invariant proportion

As for the notion of invariant proportion, it is easily understood by pi (if defined by circle's circumference\diameter ratio), which is irreducible into . AND not extensible into straight _

This invariant proportion is found among a collection of many circles, but since a collection is a composed thing, it has at most potential infinitely many circles, exactly because actual infinity is the non-composed property of at least 1-dim thing.

By ACTUALLY USING visual_spatial AND verbal_symbolic reasoning, one gets the following hierarchy of existence:

1) Actual infinity (al least the non-composed _______ thing)

2) Finite (the finitely composed that is finitely weaker than the non-composed _______ thing and therefore can be added up to a given value (for example 1, that is defined at the edge of 0______1)

3)Potential infinity (the infinitely composed that is infinitely weaker than the non-composed _______ thing and therefore can't be added up to a given value (for example 1, that is defined at the edge of 0______1)
 
Last edited:
Status
Not open for further replies.

Back
Top Bottom