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So it is logically observed that no matter how many times the radix point is "pushed" "downward" along the unbounded logical tree, no logical path "above" the radix point has an unbounded number of bits, which logically means that no amount of bounded logical paths (which are equivalent to collection of natural numbers) is infinite (or unbounded).

By this "direct" logical observation it is realized that there is a straightforward logical linkage between the common property of being logically bounded (as observed among natural numbers, as constructed along the unbounded logical tree) and the logical observation that there is no infinite (or unbounded) collection of bounded paths.

Moreover, if one observes some distinct unbounded path (which is not path 000...) as a measurement value (one logically defines number > 0 without any radix point along it) for the amount of natural numbers (as logically constructed here) one discovers that there are unbounded alternatives to such measurement value (it means that the notion of aleph

Furthermore, being uncountable is based on notions like aleph

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If one defines number only in case that there is a radix point along any given unbounded logical path, then one logically observes, for example, The Axiom of Infinity, as follows:

The Axiom of Infinity (as written in Wikipedia):

"There is a set I (the set which is postulated to be infinite), such that the empty set is a member of I AND such that for any x that is a member of I, the set formed by taking the union of x with its singleton {x}, is also a member of I."

By using radix point in order to construct natural numbers (as logically observed here along an unbounded logical tree) one logically realizes that this axiom simply "pushes" the radix point "downward" along the unbounded logical tree, and since no member of that set (which is defined by this mathematical induction) has unbounded bits, this collection has no more than finitely many members (where one of the particular cases of mathematical induction is a set of natural numbers).

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By this "direct" logical observation it is realized that there is a straightforward logical linkage between the common property of being logically bounded (as observed among natural numbers, as constructed along the unbounded logical tree) and the logical observation that there is no infinite (or unbounded) collection of bounded paths.

Moreover, if one observes some distinct unbounded path (which is not path 000...) as a measurement value (one logically defines number > 0 without any radix point along it) for the amount of natural numbers (as logically constructed here) one discovers that there are unbounded alternatives to such measurement value (it means that the notion of aleph

_{0}as the one and only one alternative, is logically insufficient).Furthermore, being uncountable is based on notions like aleph

_{0}, but since there is no one and only one alternative for the measurement value of the amount of natural numbers (in case that one logically defines number > 0 without any radix point along it), the notion of being uncountable logically does not hold (without aleph_{0}as the one and only measurement value of the amount of natural numbers, values like 2^{aleph0}have no accurate logical basis).--------------------------------

If one defines number only in case that there is a radix point along any given unbounded logical path, then one logically observes, for example, The Axiom of Infinity, as follows:

The Axiom of Infinity (as written in Wikipedia):

"There is a set I (the set which is postulated to be infinite), such that the empty set is a member of I AND such that for any x that is a member of I, the set formed by taking the union of x with its singleton {x}, is also a member of I."

By using radix point in order to construct natural numbers (as logically observed here along an unbounded logical tree) one logically realizes that this axiom simply "pushes" the radix point "downward" along the unbounded logical tree, and since no member of that set (which is defined by this mathematical induction) has unbounded bits, this collection has no more than finitely many members (where one of the particular cases of mathematical induction is a set of natural numbers).

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**So my question is this: can one please find logical failure(s) in my arguments?**
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