Epsilon = Invariant Proportion
About 3.14... = circumference/diameter:
Let us say that Epsilon is equivalent to the invariant proportion that can be found in the triangles below.
(VERY IMPORTANT:
When Epsilon = Invariant Proportion, then there is no connection to words like 'smaller' or 'bigger' or 'size' or 'magnitude' or 'Quantity', and the reason is clearly explained)
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|____|____|___|__|_\
Each arbitrary right triangle's area is smaller than any arbitrary left triangle's area, but the internal proportion of each triangle remains unchanged, so it does not depend on size or magnitude (please think about circumference/diameter ratio, which does not depend on a circle's size).
If we have finitely many triangles then this proportion can be found finitely many times.
But in the case of infinitely many triangles, this proportion can be found infinitely many times.
Since Epsilon is equivalent to this proportion, it cannot be found if and only if this proportion cannot be found.
It is clear that if the proportion can be found infinitely many times, than it cannot be eliminated, and if it is eliminated, it means that it is found only finitely many times.
In other words, any collection of infinitely many elements can be found if and only if some epsilon that belongs to it also can be found, and if this Epsilon cannot be found, then there are only two options, which are:
a) The collection does not exist.
b) The collection is a finite collection.
Conclusion:
There is an inseparable connection between the PERMANENT EXISTENCE of an epsilon and the collection of infinitely many elements that is related to it.
In other words, there is no way to calculate the exact SUM of infinitely many elements, because the SUM of infinitely many elements cannot be more than SUM – epsilon, and therefore the accurate SUM of infinitely many elements does not exist.
Therefore 3.14... < The accurate value of circumference/diameter.
About |N|:
The idea of Epsilon = An invariant proportion, is not limited only to a collection that can be found on infinitely many different scale levels.
In other words, we can use this idea in order to show that the accurate value of |N| is undefined by definition, where the definition is not else then the ZF Axiom of Infinity, for example:
, , , , ,
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| \ | \ | \ | \ | \
| [i][b]1[/b][/i] \ | [i][b]2[/b][/i] \ | [i][b]3[/b][/i] \ |...\ | [i][b]n[/b][/i] \ [i][b]n[/b][/i]+1
|____\|____\|____\|____\|____\ ... ad infinitum.
In this case Epsilon = 1, but then we can clearly see the mistake of Cantor's approach, because if n+Epsilon is in N (by the ZF Axiom of Infinity), then the accurate value of N is undefined because we have a permanent state of |N| - Epsilon.
What to you think?
