Infinity: Old News?

The Infinity Illusion
I recently read an article in New Scientist that got me thinking about one of the ideas that make up the very foundation of mathematics. “We know we can’t reach it in a day, year, decade, but we know it’s out there, and if there were a hundred, thousand of us all traveling for…well…infinity, we can certainly reach it!” Now before you mathematicians and non-mathematicians alike begin yelling at me for doubting such a grand mathematical concept as infinity. (“How dare you make light of a concept that has been around for thousands of years, a concept that has been engrained in our minds since our earliest childhood memories?–Oops, little Timmy, you can’t find the slope of that curve. It’s infinite! ) I’d first like to say that I have always been fascinated by the concept of infinity (a look at the header photo of my blog shows just that), so I personally do not believe that we can just throw away centuries worth of mathematical practice. Secondly, I am only talking about infinity in the mathematical sense and with the intention of explaining all sides of the argument. I leave off all arguments that do not pertain to math.

With that said, let’s embark on a journey through the wonderful world of infinity.

Forget everything you thought you knew about mathematics: there is a largest number. Start at 1 and just keep on counting and eventually you will hit a number you cannot exceed-a kind of speed of light for mathematics.

If you read the quote several times, you might starting thinking that representing infinity as a really large number doesn’t sound that bad. Computers (classical computers, I should add) are limited by the number of bits in their memory. Due to physical constraints, the computer’s processor has a bound on the largest number that can be represented. Going above that largest possible number results in overflow and can potential cause costly disasters to occur.

What is represented as infinity in Java, for example, is

Double.POSITIVE_INFINITY

and

Double.NEGATIVE_INFINITY

. Don’t get fooled by the inf that prints out when you do System.out.print. These numbers are actually represented by a finite number of bits, more precisely the value returned by

Double.longBitsToDouble(0x7ff0000000000000L)

. A really large number but not quite infinity.

So you wonder: if computers can get along just fine without the need for anything resembling real infinity, then why do we humans need to have this abstract concept that seems to have no practical use. Although as much of a pain infinity can be in various fields (to see a list, consult the article), we can’t rule out a concept simply by pointing at current technologies. Just because we can’t handle infinity on the hardware we currently have does not mean that the technology of the future theoretically cannot. (In fact, by the looks of it, quantum computers may come close to juggling near infinity in the future.)

Before going any further in exploring various mathematicians’ and scientists’ opinions on infinity, I need to make a distinction between the two types of infinity that mathematicians consider. The first is actual infinity which is a total complete set composed of all its members including infinity. The second is potential infinity which is an incomplete set where more members can always be added to the set, possibly to an infinite end. Jeremy Gwiazda recently proposed a method to distinguish between the two types of infinity in his paper, “Throwing Darts, Time, and the Infinite”.

If something is actually infinite, then selections do not grow through time. So
by the contrapositive, if selections grow through time, then we are dealing with a potential infinity.

Let’s imagine that you have a number line in front of you numbered with the natural numbers {0, 1, 2,…}. If you throw darts at the infinite number line, the first dart will land at a whole, real number. The second dart, by Gwiazda’s method, would land at a number that is larger than the number that the first dart landed on. The rationale behind this is that there is an infinite number of points beyond the point that the most recent dart just landed on. For example, if the first dart landed on 100, then there are an infinite number of choices to choose from for the second dart to land on that is greater than 100. There is only a finite set of choices for the dart to land on less than 100. Therefore, the second dart is bound to land at a number higher than 100. If this is the case, you are looking at potential infinity because your dart selections increase over time.

This method certainly works for the case of the natural numbers above and the paper makes an interesting and novel quantitative observation on the behavior of potential infinity. But it is also important to consider scenarios other than the natural numbers, where potential infinity also exists. For cases in which we consider negative numbers, such as the infinite set of all integers, it would be logical to state that the absolute value of selections over time grows. This is due to the fact that only the distance from the previous dart hit to 0 is finite, and infinities bound either side of this stretch of finite length.

Sad-Face
In the case of all the natural numbers, if a dart lands at 1 in the first throw, the dart can either land between 0 and 1, land at a real number greater than 1, or land at a number less than 0 with equal probability (at least not with a probability of one option greater than another). This is the case because the number of reals in \left\{0,...,1\right\} is infinite, the set \left\{1...\right\} is also infinite, and finally (you guessed it) the set \left\{x:x<0 | x \epsilon \mathbb{R}\right\} is also infinite. We cannot compare whether one infinity is “larger” than another infinity and, thus, cannot say that selections will grow over time (since the next dart can very well land between 0 and 1), but potential infinity still exists.

I did not introduce the two infinities to spur a philosophical argument about whether the infinite set of all real numbers between 0 and 1 is a potential infinite or an actual infinite (if that idea was ever spurred at all!). I did introduce them to better explain various mathematicians’ and philosophers’ views on the existence of infinity. While most mathematicians believe that potential infinity exists (except for the few who think that even the concept of infinity is chimerical), the existence of actual infinity is what we debate when we say that infinity does or does not exist. (Remember? That’s what we were talking about in the first place.)

I see it, but I don’t believe it!

Georg Cantor was one of the first to show the existence of what he believed to be actual infinity. Cantor defined a type of “countable infinity” in which numbers in an infinite set are mapped to the natural numbers. For example, consider the set of integers greater than 100: \left\{101, 102, ...\right\}. You can map this set to the natural numbers as such:

1 \rightarrow 101

2 \rightarrow 102

3 \rightarrow 103

If an infinite set can be mapped to the natural numbers in this way, then the set has the same “size” or cardinality as the set of natural numbers (this cardinality Cantor denoted as \aleph_0). Cantor went on to show that the set of rational numbers can also be mapped to the natural numbers and thus also has cardinality \aleph_0. However, the set of all real numbers cannot be mapped in the same way. And Cantor explains it as thus: If you have a set of real numbers composed of an infinite number of integers, then you can always create a number that is different from every other number in the set by arranging the numbers in a rectangle and changing the first digit of the first number, the second digit of the second number, and so on. To see an example, let consider the following set of real numbers between 0 and 1.

0. 1 111111111...
0.1 2 34567899...
0.22 3 4343434...
0.500 0 000000...
0.3333 3 33333...
0.77777 7 7777...
0.250000 0 000...
0.9999999 9 99...
0.34455888 8 8...
0.125600000 0 ...

0.2341461091...

If we add one to every number in the diagonal (highlighted red), then we can create another number (highlighted blue) that is different from every number that is already listed in the set. Given that we cannot list all real numbers, we cannot also map the set of real numbers to the set of natural numbers.

Cantor’s result suggests that “different” infinities exist and that the “infinity” of the set of real numbers is larger than the infinity of the set of natural numbers. Cantor defined \mathbb{R} as the continuum with a cardinality of \mathfrak{c} where \mathfrak{c} > \aleph_0. Since one particular infinite set can be greater than another infinite set, we can almost imagine a sequence of infinite sets, each “larger” than the previous. In fact, you can create this sequence of ever greater cardinalities by taking the power sets of infinite sets and further taking the power sets of the power sets of the infinite sets. (Whew!) (Cantor proved that the power set of a set has greater cardinality than the original set.) The set of ever larger “infinities” contains within itself the infinite sets of numbers. The existence of infinite sets within infinite sets points to actual infinity because the infinite sets that compose the larger infinite set are self-contained.

cookies

If that sounds like a bunch of math mumbo jumbo, consider this: A chocolate chip cookie contains a set of chocolate chips, and a cookie jar contains a set of cookies. If there are an infinite number of cookies in the cookie jar, each cookie still contains all its chocolate chips and is still a complete cookie even though the number of chips might be infinite.—-If there were no cookie jar, then a cookie containing an infinite number of chips will not be a complete set because you do not know if there exists other cookies. You only see one endless cookie. (Not such a terrible thing though if I may say.)

Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. Therefore we must conclude that w is not a predicate.

But Cantor’s work also contains characteristics that lead to contradictions like Russell’s Paradox. Let R be the set of all sets that are not members of themselves. If R does not contain itself, then by definition it must be part of the set of all sets that are not members of themselves. In other words R must contain itself. But by this definition, we have reached a paradox, R cannot simultaneously contain itself and not contain itself.

let\,\, R = \left\{x|x \notin x\right\},\,\,then\,\,R\in R \Leftrightarrow R\notin R

To resolve such a paradox, some set theorists propose that the rules of set theory should be applied in most but not all cases, while others propose new systems such as the ZFC axioms.

Yet, another branch of mathematicians choses to deny the existence of infinity in mathematics. These finitists believe only in the existence of finite mathematical objects and any infinite object such as an infinite set does not exist. For example, finitists believe that all natural numbers exists, but the infinite set that contains these finite numbers does not.

When you boil down to it, whether the concept of infinity is useful in mathematics is a question of notation. We live in a world where the finitists are greatly outnumbered by their Platonic counterparts. That doesn’t mean that finitists are restricted by their belief that actual infinities do not exist. A theorist can choose to accept theories that support their beliefs and reject those that don’t.

As for the question of whether infinity exists, how would we know? The basic concept of infinity is that we can’t reach it by any of our worldly methods (if we can reach it, then whatever we’ve reached isn’t infinity). But I’m sure that every middle schooler would be happily content with know that infinity is one their side by presenting the option to throw away any limit problems because— little Timmy whines, “I can’t calculate the limit since the function increases to infinity!”

As for knowing the secrets of infinity…well, what can you expect, we’re only human.

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4 Responses to Infinity: Old News?

  1. David Zhang says:

    “You only see one endless cookie. (Not such a terrible thing though if I may say.)”
    So good.

  2. Thank you for discussing “Throwing Darts, Time, and the Infinite.” You make some interesting points and the paper perhaps was not as clear as it should have been. You quote two important sentences:

    “If something is actually infinite, then selections do not grow through time. So
    by the contrapositive, if selections grow through time, then we are dealing with a potential infinity.”

    From this, however, it does not follow that “If selections do not grow through time, then it is actually infinite,” which I take to be your main counterexamples. In this paper, I was really focusing on arguing that sequential random selections’ growing through time implies a potential infinity. And I think that this (being merely potentially infinite) is the best explanation as to why such selections do (at least seem to) grow through time from the positive integers and a well-ordering of the reals. A start towards the work of arguing that a hyperinteger is an example of the actual infinite, and is in fact the correct extension of the concept of finite whole number into the infinite, can be downloaded here:

    http://philpapers.org/rec/GWIOIN

    • Quanquan Liu says:

      Oh yes. It seems I’ve made an oversight and an overgeneralization error in my interpretation of your paper. Apologies for the mistake. I have fixed my blog post to account for it. If I understand it correctly now, I was making the assumption that if selections do not grow through time, then it is actually infinite (or in other words, it cannot be potentially infinite). But I see in your paper that you do not preclude that possibility. You are simply making the observation that if selections grow through time, then we are dealing with potential infinity. It, however, doesn’t preclude the cases where selections might not grow through time, but we see potential infinity (as in the case of the real number line)?

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