Infinity - Sizes vs Types

This is already acknowledged in that we can single out primes, evens, odds, algebraic, transcendental, complex, etc. numbers. These are different types (not in the strictest sense) of countable infinity.

By contrast, waiving an overly intuitive argument from the appearance of lines being divided variously, let us imagine trying to count a set of real numbers from 0. Now each real number has countably many digits. Thus if we count from 0 to some "next" number, say the first number with a nonzero digit in its millionth decimal place, we will be skipping over all the numbers with first nonzero digits in their post-millionth decimal place (which are fractionally smaller than the number we skipped to). And so on and on, and this whole uncountable ocean of decimal expansions is reiterated from 1 to 2, 2 to 3, and so on.

On a more abstract level, we have to track the distinction between countability and iterability. They seem similar; and along your lines, we also know that every definable set of reals is quasi-countable vs. the image of the set of all countable ordinals (omega1). Sometimes this even collapses our image fully back into countable infinity. (This is not the same as, but relevant to, of course, things like Skolem's paradox.)

Great question, I have been studying this for some years.

Here are a few short thoughts:

There are alot of issues when dealing with infinity and ironically it can get chaotic as we are dealing with fundamentally questions in measurement that are not just roots in counting and mathematics but also how we qualify phenomenon rooted in a very basic platonic form.

1. Axiomization of space is infinite in nature. The line and line segment are barely seperated outside of the mathematical community. Some will say a line is two directions, stemming from a center point in two directions continuously. Others will say it is strictly a 1 directional line between two points. Others will say it is strictly a line between points and forget direction altogether. Other's will say a line (segment) is composed of an infinite number of 0d points. Other's a line is composed of an infinite number of lines, as you cannot quantify zero without equating it to 1.

It is the axiomization of this spatial form itself, that requires various linear tautologies of reason to define what it is or is not. And we are left with communities, professional and not (or of various proffessions) disagreeing over its nature.

2. Using the premise of a simple 1 directional line between two points:

A. The line is both a generality and a particular or one and many, and we are left with a paradox.

If I have a standard 1 directional number line, let's say going from 1 to 3, it is still one number line. However it is composed of 3 lines. It is 1 line composed of 3 lines. It is a general set of lines and 3 particular lines. Particular on the respect each of the 3 lines is a "part" of a whole.

Now if I take each of those particular lines and divided it into 3 lines, those particular lines are now composed of particulars so they now exist as sets of lines or as "generals". These lines are both particulars and generals.

Because a line is a line, every time you divide the line into parts you also multiply it considering a line is a line is a line. What differs one line from another is its size or "ratio". How many times does one line fit inside of another. Thus the ratio is self referential: the line is measured through the line. It is circular reasoning.

The line as a general and a particular reflects the basic measurement problem in philosophy of the "one" and the "many" where the one and many are strictly United under a single third term of "form" (line in this case).

And it get further complicating as if you take each line as both as general set and a particular and continue multiplying/dividing it we have a paradox. If each line is continually divided by three, the original general line (and each other general line by default) manifests an infinite number of lines. This occurs of you also take the original line and multiply it by 3 continually.

So the line is both a general set and particular that "linearly" is manifests more and more lines through time. But this is linear as well.

3. Each line as both composed of and composes further lines is an infinity. Thus we we are looking at one line we are looking at an infinite set and we are quantifying an infinity. Finiteness thus, in this example, can be observed as multiple infinities.

4. The line (segment) is circular in nature. It has the same beginning and end points (ie the 0d point). You cannot say that one point is different from a other as that would require you quantifying 0 and equating it to 1. The nature of the line measured through the line, as lengths, requires a self referentiality as well and this is circular....thus ratios are circular reasoning by nature.

5. HERE IS ANOTHER INTERESTING PARADOX: Each line (segment) is of 0 width. If you stack one line one top of another line, with no linear distance between them, you are stacking zero width objects on top of eachother. If you have a line between them it is also not just 0 with but isomorphic.

________A

B

________B

If line A is 1 dimensional and line B is zero dimensional, and both have zero width, and line B is a line between them then line be is a 0d line. This is a contradiction in standard mathematical reasoning as you have a 0 dimensional line that seperates one line from another line no different than a 0d point seperated one line from another on a number line.

Thus a single line, as having zero width, can composed of multiple lines stack on top of eachother and still appear as one line.

Now you have:

______A

B

______C

D

______E

As

______(A,C,E)

If lines A C and E are each divided into 3 lines we are left with the line being a condensed matrix.

6. Quantity and Quality. The line begins with a 0d point. The 0d point negates itself into another 0d point, and we are left with the first quality of length. A single line, measured against no other line is always infinite as there are not comparisons. It is a quality. Only when the line is multiplied/divided into another line does quantity result. Thus quality has a nature of generality and quantity finiteness. A quality is an infinity (no different than a quality such as color being composed of infinite colors) and a quantity is multiple qualities.

7. All counting is grounding in form. We count forms. The line (segment) is the most basic form. If I count a line I am equating a form to a number.

This form, because it has both the same beginning and end point, is circular in nature. (Imagine watching the circular movements of a clock hand from the side and you have a line going back and forth).

The counting of the object is also circular as it is a loop between the subject and object. Where the assumptive formless nature of the subject Inverts an object (form) into another object (form) by dividing/multiplying it.

Numbers as counting phenomenon, with the line being the most basic, are inseperable from forms. This forms are various loops as the line itself with both the same beginning and end points are tautologies, thus loops.

The number is thus inseperable from form in both form and function and is self-referential as considering it is a form is "countable". We can count numbers as counting is the creation of form.

8. The lines as ratios of other lines, result in lengths. One infinity is thus larger and smaller than another infinity, yet both are infinity as both are lines. Types are merely tautologies and tautologies are ratios of one underlying axiom self referencing itself.

What you're suggesting is that we can slice an infinite real line up into chunks. We number them "chunk 1", "chunk 2" and so on to create a countable representation of the real number line that completely covers it. Huzzah, the real line is "countable". However, because we're talking about the Real number line, you can use a kind of Epsilon-Delta move to move the boundary of any given chunk a little bit further up or down the line while still having your countable set of chunks be a total cover. So, yes, your representation has countably many sets that cover the line, but you can also prove (thanks to what we know about real analysis) that there are uncountably many such representations.

I'm still getting to grips with algebraic topology, but I think there is an important distinction to draw here between whether the ability to interpret prima-facie transfinite topologies like real geometry in a combinatorial way implies that one can do without transfinite methods and whether this implies that there are no transfinite infinities. In a brute force, platonistic kind of way, we might say that the former is an entirely sensible position to take without having to draw any kind of conclusions about the latter. Combinatorial mathematics could surely work with really interesting tools discovered through deep dives into the transfinite without themselves needing explanatory tools beyond the rational.

Basically, we know that there are uncountably many such divisions of the real line into countable sets. And, in fact, we know that we need transfinite mathematical methods in order to really understand how such divisions are constituted and relate to one another. That's metamathematics. Actually doing the work we want to do may well be possible on entirely rational-valued metric spaces using finite operations in computable time, which is great, and with our practical engineers' hats on we might be entirely uninterested in the difference an epsilon makes; but understanding our metamathematics helps us get there.