# Can infinity be made finite in certain conditions?

In mathematics there are not only infinitely big numbers, but also infinitely small numbers. One can consider arbitrarily small numbers that can exist only in the mathematical world. For example, ten to the power minus one billion, zillion ... whatever you can think of.

That is why if you just consider a finite portion of the number line, like from zero to one for example, you can find infinitely many numbers between zero and one. But that particular portion of the number line is finite.

Now consider this example. Suppose you are given an infinite number of bricks which are infinitely small in size. If you construct a building using all of those bricks (it seems impossible though, but this is philosophy), will the size or the volume of the building be infinite or finite?

• See Continuity and Infinitesimals and see Infinitesimals. Oct 29 '18 at 8:32
• In the real world there exist a minimum length, the Planck length, which would make an actual construction impossible. As a pure math construct, you may well find different answers in various mathematical frameworks. Oct 29 '18 at 9:34
• There are no infinitely small numbers in standard mathematics. Your example of 10 to the minus a bazillion is a positive real number, and the interval between 0 and 10 to the minus bazillion is topologically identical to the entire real line, just as the unit interval is. Oct 29 '18 at 9:38
• I made an edit which you may roll back or continue editing. Welcome to this SE! Oct 29 '18 at 11:43
• I am neither philosopher nor mathematician but surely the only infinitely small number is zero. If the bricks have 0 dimensions you would never build a house even with an infinite number of them. Oct 29 '18 at 13:16

In modern mathematics here are not really infinitely small numbers. Analysis was carefully reformulated to remove them. There are models that involve ideal elements, but that is not really the same thing.

Your ordinary mathematician no longer uses infinitesimals, or even literal infinities, they use formalisms that involve limit processes, instead. For instance, something is true at infinity, in modern analysis, if whenever it is true of everything larger than a given size. The phrase 'is true at infinity' is just translated into that notion of an unbounded sliding scale and done away with as a genuine concept.

But even among those who do use them, the notions of infinitely small and infinitely large are not complete on their own. Of the infinitely large, there is more than one size, and the most relevant one of those sizes comes in several varieties. Of the infinitely small, there are also several varieties that all act differently.

So the question has every answer you might want. If your infinitely small bricks are point-sized, and you have countably many of them, your wall is definitely "of measure zero". If you have uncountably many of them, it can be any size you want, and you can keep pulling walls the same size at it out of it infinitely many times without making it any smaller or less dense. (This is the Banach-Tarski Paradox.) So we generally avoid collections of things that are point-sized, other than points, because that makes no sense.

But if they are infinitesimal (from a Nonstandard continuum) but larger than points, we have to be very careful about how we handle them, and we can create an even larger infinitude of sizes, including zero.

The question is Can infinity be made finite in certain conditions?

The OP provided an example where this might be possible:

That is why if you just consider a finite portion of the number line, like from zero to one for example, you can find infinitely many numbers between zero and one. But that particular portion of the number line is finite.

However, the question remains in this further context:

Suppose you are given an infinite number of bricks which are infinitely small in size. If you construct a building using all of those bricks (it seems impossible though, but this is philosophy), will the size or the volume of the building be infinite or finite?

The OP also considers "arbitrarily small numbers" as numbers like "ten to the power minus one billion, zillion ... whatever you can think of". I will assume this idea of arbitrarily small is what is meant by "infinitely small" in this question.

If that is the case, then a convergent series may be an example of what is desired. This is Wikipedia's description of a convergent series:

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

A series is convergent if the sequence of its partial sums {S1, S2, S3, ...} tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases.

Not all series of infinite sequences of numbers converge. An infinite sequence of the number 1 would not converge as a series. However, if the infinite series did converge, this may be one way to get what the OP is looking for.

Reference

Wikipedia, "Convergent series" https://en.wikipedia.org/wiki/Convergent_series

• The infinite sequence 1, 1, 1, 1, ... converges to 1. Surely you know this. Did I misread you? Ah you are confusing the terms series and sequences. You need to review those terms. The sequence 1, 1, 1, 1, ... converges; the series 1 + 1 + 1 + ... doesn't. Oct 29 '18 at 18:00
• @user4894 Good point! I think I corrected it. Thank you for pointing that out. Oct 29 '18 at 20:34

The difference between what is infinite and what is finite is that what is finite exists for our perception. What is infinite does not. We cannot interact with infinite stuff.

So, you cannot build a thing with an infinite number of bricks, whether they are small or big. Because that doesn't exist. If we can say anything, it is that the result will not exist.

..."it seems impossible though, but this is philosophy". A flawed proposition cannot be the base for a logical construct.

• Aren't there quantities that are finite yet far beyond our perception? There are only 10^78 hydrogen atoms in the observable universe. That's a relatively small finite number. Oct 29 '18 at 22:37
• @user4894: the statement is: what is infinite does not exist for our perception. You apply the fallacy "affirming the consequent" and imply that if it does not exist for our perception, then it is must be infinite. Wrong. The fact that "it is far beyond our perception" or that we are unable to interact with it does not imply that it is infinite. The number you refer to is finite and escapes our perception. Oct 30 '18 at 2:07
• But I am quoting DIRECTLY from your words: "what is finite exists for our perception." I reiterate my point. Plenty of things are finite yet can never be perceived. A set containing 10^2000 hydrogen atoms for example. Surely you are not denying what you wrote and what's plainly there for all to see. "The difference between what is infinite and what is finite is that what is finite exists for our perception." Do you have a cat? My cat likes to post in my name. Thinks he's clever. Oct 30 '18 at 3:11

The question can be answered using basic geometry and its axioms.

1. The line is infinite.

2. Divide the line into x number of lines where each line segment is now both x number of lines and 1/x a line.

3. Each line is infinite, and the line of which they compose it also infinite.

4. Finiteness is multiple infinities.