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Old 19th January 2019, 02:22 PM   #3281
doronshadmi
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Wholeness is not necessarily Comleteness

Wholeness is not necessarily Completeness, as seen in http://www.internationalskeptics.com...postcount=2798 exactly because infinitely many things are infinitely weaker that actual infinity (as seen in http://www.internationalskeptics.com...postcount=3095).

In order to deal with such notions, philosophy and mathematics are inseparable of each other (http://www.internationalskeptics.com...postcount=3280).
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Last edited by doronshadmi; 19th January 2019 at 02:46 PM.
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Old 29th January 2019, 04:36 AM   #3282
doronshadmi
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Please look at the following diagram:



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.
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Last edited by doronshadmi; 29th January 2019 at 06:23 AM.
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Old 29th January 2019, 07:21 AM   #3283
doronshadmi
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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.
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Old 29th January 2019, 07:30 PM   #3284
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Old 31st January 2019, 08:30 AM   #3285
doronshadmi
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The standard notion of set (according to Prof. Melvin Randall Holmes):
Quote:
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}}


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.
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Last edited by doronshadmi; 31st January 2019 at 08:50 AM.
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Old 2nd February 2019, 11:52 PM   #3286
doronshadmi
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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 ).
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Last edited by doronshadmi; 3rd February 2019 at 12:11 AM.
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