I wrote a paper thats explains how an infinite number of points constitutes a finite space and that infinite spaces are mathematically undefined.
I made some changes in the results section and included a second version.
The third paper is even more concise in formula and explanation.
\(n/(x-1)\) is an extremely simple function that super-smart students don't bother to take its limit (it's not worth their effort). But it models a real physical phenomena and therefore gives results as far as to defining infinite space's nature.
You don't have to close the gaps with \(x-1\). You can fully close them with any way you want with any function with limits. I also used \(2^x\) to close the gaps. The trick here is to make gaps get equally smaller. All gaps have to have equal length while getting smaller equally.
Two animated illustrations above are examples of how a space begins to form as points evenly distributed on the line increase. You see towards the end that a space will begin to form, and this is only true when space has a finite length. Intuitively, you can never fill gaps in an infinite length; the gap length between two points would be undefined. The illustrations are consistent with what is shown in the paper. You cannot form an infinite space given the fact that a space is composed of points — indivisible units.
Since space is finite, coordinate system is finite. Here is the confusing part though, when x-axis is finite, limits should not reach infinity (there is no graph that extends infinitely where the limit can reach infinity), which makes building a continuous finite space not possible. But this is a paradox. You need to reach infinity to build a continuous finite space but that, in turn, means limits cannot approach infinity. However, it would theoretically be possible to build a space that is not continuous in case limits do not approach infinity.
Finite space construction isn't the only proof that outer space is finite. The other proof is much simpler and requires less rigorous analysis. You can multiply a finite line (a finite space) with any real number and the result is finite but not with infinity and that is the answer.
Former teachings of mathematical spaces taught us that space can be both finite and infinite. For example, they said the real number line extends infinitely in both directions while constituting an infinite space. I particularly want to falsify this statement. I earlier proved an infinite number of points make up a finite line. Therefore, if a line extends infinitely in one or both directions, that means that it cannot consist of points. But if a line consists of points— which we know is the case— then a line cannot extend infinitely in one or both directions. Since we know for sure that a line consists of points, then Euclidean space and thus outer space is finite.
Outer space has all properties of 3-D Euclidean space. I earlier proved that Euclidean space also has the finiteness property. Can we then say that outer space satisfies this property as well? If so, outer space as we perceive it is finite.
Even if you make the assumption that infinity as a concept exists (limits reaching infinity), spaces of all dimensions are still finite. If you accept that infinity doesn't exist, then spaces are finite anyway. We can conclude that mathematical infinity is not physically real; however, it remains a useful abstraction.
I would like to add one more thing: if a finite number of points constituted a continuous finite space, then we could say infinite space exists.
If we leave aside the effects of mass and energy described by Einstein's general theory of relativity, outer space satisfies all properties of Euclidean spaces. I have proved that Euclidean space's another property is that it must be finite. Could you then say for sure that outer space satisfies Euclidean space's finiteness property? If so, outer space as we perceive it is finite.
We know for sure that outer space satisfies all properties of Euclidean spaces except for finiteness property. Does the fact that outer space satisfies all properties of Euclidean spaces while leaving aside finiteness property mean that it would also have to satisfy finiteness property?
The answer to the question whether outer space satisfies Euclidean space's finiteness property is that if outer space has a notion of point like in mathematics, then it satisfies the finiteness property, meaning that it is finite. Outer space being finite is tied to the fact that it is composed of indivisible units — in space's context, it is a mathematical point. Indivisible units are what make space finite.
Note: Originally, my paper replaced n, which is the length of the line, with the infinity sign to indicate infinite length. However, generative AIs (Gemini, ChatGPT) say this isn't a correct operation because infinity is not a defined value that can be substituted into n -- which is the length of the space. I write my new equation here:
\[ \lim_{\substack{n\to \infty \\ x\to \infty}} \frac{n}{x-1} = \frac{\infty}{\infty} \]
The thing is, \(x-1\) is not important. You can remove -1. It is just there to represent the exact relationship between the number of points and gaps. Then you have:
\[ \lim_{\substack{n\to \infty \\ x\to \infty}} \frac{n}{x} = \frac{\infty}{\infty} \]
which is a more concise representation of the same concept. I need to recall that this is exactly the same as what Riemann sums say in regards to how rectangles approach infinity. I just add the detail that the interval over which rectangles become infinite in number approaches infinity while proving that it is undefined.
\[ \lim_{x\to \infty} \frac{n}{x} = 0 \]
When n is a constant, meaning that the interval is finite, the limit is well defined and equals zero, meaning that gaps between all points are fully closed while forming a continuous finite space. The limit is proved to not approach infinity even if we make the assumption that it does. But that means that the space should not have formed in the first place. Then how does outer space exist while being both continuous and finite? I think this is an interesting question.
I answered the philosophical question that if you can get infinitely smaller. The answer to this question is
I sometimes open a ncat server on ssl on
You will run the following command: ncat --proxy 127.0.0.1:9050 --proxy-type socks5 2gk5ec5ejjqnl6b67fxaceftsl5nd2qwfjtwqut7trgz3wqjx6awntqd.onion 80 --ssl