Is There a Logical Limit to Infinite Collections in Unbounded Logical Trees?

[Look]

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 aleph₀ as the one and only one alternative, is logically insufficient).

Furthermore, being uncountable is based on notions like aleph₀, 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 aleph₀ as the one and only measurement value of the amount of natural numbers, values like 2^aleph₀ have no accurate logical basis).

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

 

[mmwave]

I would first like to commend you for your thorough and logical observations. However, I do see a potential logical flaw in your argument.

You state that there is no infinite or unbounded collection of bounded paths, and therefore, the notion of being uncountable does not hold. However, this may not necessarily be true. It is possible for a collection to be uncountable even if it is made up of bounded paths. This is because the concept of countability is not solely based on the number of paths, but also on the relationship between those paths and the elements they represent.

For example, the set of real numbers is uncountable, even though each real number can be represented by a bounded decimal expansion. This is because there is no one-to-one correspondence between the real numbers and the natural numbers, which is the basis for countability.

Additionally, your argument seems to rely heavily on the concept of measurement values and the use of a radix point. While these can be useful tools in mathematics, they may not be applicable to all situations and may not accurately represent the properties of a given set.

In conclusion, while your observations are valid and thought-provoking, I think it is important to consider the limitations of using certain mathematical concepts and tools to make broad statements about the properties of sets. Further research and analysis may be needed to fully support your argument.