## How can the finite contain the infinite?

1

This article claims that all finite things are actually infinite, however it doesn't explain how this can possibly work only that it is the way the world really works. Numbers and lines are finite, but they can undoubtedly be divided infinitely, thus the finite ultimately contains the infinite!

But that is the most illogical thing and seems to me contradictory, how can something be finite and infinite at the same time time (perhaps this is essentially Zeno's paradox)? Isn't the infinite supposed to be endless and boundless, which is the exact opposite of the measurable and the quantifiable?

So my question is twofold:

1. is it true that the finite contains the infinite (whatever that means)?
2. how can it be explained logically?

2Numbers and lines do not "really" exist, they are idealizations, mathematical questions about them do not concern how the world "really" works. Something can be easily finite and infinite at the same time in different senses, as here, finite in size, infinite in the number of parts. A sphere is endless and boundless but not infinite in size, and a segment has bounds and finite size but contains infinitely many points. – Conifold – 2018-03-18T22:08:26.440

1That article is crap (that's a technical term). It doesn't distinguish between physical reality and abstract math. A common error in these Zeno discussions. – user4894 – 2018-03-18T23:43:40.213

you might like to read about the Mandelbrot set and fractals.... – Swami Vishwananda – 2018-03-19T04:52:54.967

@user4894 are you saying that in reality things are finite but in theory they can be infinite? I don't see how that makes sense either. – Bach – 2018-03-19T13:58:43.853

According to our best contemporary physical theories, the universe and its contents are finite. Mathematics routinely deals with the infinite. It's an important distinction. The mathematical solution to Zeno's arrow paradox depends on infinite divisibility of space and time. In physics, the Planck scale restricts what we can sensibly say or know below certain intervals of space and time. – user4894 – 2018-03-19T16:45:12.860

One has to be careful with what is meant by "finite" for the question to make sense "in reality". For example, it is meaningless to ask if "real" space has infinitely many points (or if "real" fields are continuous), those are properties of mathematical models, not the reality itself. One can create equally good models based on either choice. – Conifold – 2018-03-21T02:08:49.657

1

A finite line segment can be viewed as the sum of infinitely many smaller line segments that don’t overlap by using the technique in the article attributed to Zeno of Elea. That is, cut the line segment in half at a single point. Let that single point one used to make the cut go with the first half. Keep that first half, but cut the other in half again using the same method. If one could do that infinitely many times, which is unlikely, one would have infinitely many line segments that are disjoint and the sum of whose lengths would be equal to the same length as the original line segment.

So, one has a line segment of finite length that could be thought of as composed of infinitely many non-overlapping line segments of smaller lengths.

Alternatively, one could look at what is finite on the one hand as the cardinal number of a set containing one line segment that has a finite length, call it L. What is infinite on the other hand is the cardinal number of a different set containing smaller line segments whose sum of the lengths is also L. What ties these two sets together is that the sum of the lengths of the line segments in each of these sets is the same number, that is, L.