Arturo Villarreal Math89s

A number can invoke an array of emotions; from the gloomy cloud that follows a subpar test score to the burst of euphoria induced by a large number on a pay slip. Do large numbers always equate to happiness? Probably not, a large amount of personal debt wouldn’t necessarily make for the happiest person. Regardless, large numbers are interesting and can even challenge reality as they climb up in magnitude. What is the largest number you can think of? Wilt Chamberlain’s record of 100 points in a basketball game is a large number, but it is by no means a respectable candidate for largest number. The current national debt of the United States stands at a whopping $19 trillion; a considerably huge number but falls to the heels of a googol (10¹⁰⁰). Have we finally found a number that triumphs all? Well no matter what number you can think of, it will always fall to a plus one value. This principle means there really isn’t a biggest last number thus our question implodes through its own logic. How about infinity? It’s commonly known, at least by the general public, that infinity bests any number you can think of. However, infinity isn’t really a number and there are different kinds of infinities. As we try to journey past infinity, we must master certain principles and axioms that will be crucial to comprehending ideas past the realm of infinity.

Formulated in 1880 by Georg Cantor, the notion of cardinality has been used to compare a single aspect of finite sets. By definition, “Cardinality is defined in terms of bijective numbers (Marton, 2014).” Although the concept sounds difficult to grasp, it is quite intuitive. If two sets of things, such as basketballs and shoes, are of the same magnitude, it is said they contain the same cardinality. A number that refers to an amount of something is known as a cardinal number which is represented as a natural number (Marton, 2014). This distinction will be vital as we journey past infinity.

Let’s introduce our next concept by welcoming back a slightly modified version of our introductory question: how many natural numbers are there? The answer can’t be a natural number (n) itself as there will always be a number (n + 1) that is greater. Instead, there is a unique name for this value known as aleph-null (א‎₀) (Anderson, 2014). Aleph-null is considered the first and smallest infinity; it encompasses all rational numbers through a one to one correspondence that will be further discussed later in this paper. We can think of aleph-null as the conventional infinity we associate infinity with; the point is that aleph-null is larger than any finite amount. However, we can reach beyond this using the mechanics involved in a Supertask (Manchak, 2016).

The first steps: with a little help from Supertasks

A supertask is, “a task that consists in infinitely many component steps, but which in some sense is completed in a finite amount of time.” The term “supertask,” was coined by J.F. Thomson in 1954 but has been studied since the 5^th century B.C.E (Manchak,2016). Essentially, infinite actions occur in a finite amount of space; for our case, we can think of dividing the space between two points by two in an infinite loop. Exercising this logic, if we draw a bunch of lines and make each next line a fraction of the size and a fraction of the distance from each last line, then we can fit an unending amount of lines into a finite space.

Figure 1(Vsauce, 2016)

Since the number of lines in this space is equal to the amount of all possible natural numbers; the two can be matched one to one which gives both sets the cardinality: aleph-null. What happens if we add an addition line? One might speculate we would end up with א‎₀ + 1, but this logic is terribly flawed. Unending amounts aren’t like finite amounts; there are still only aleph-null lines since the can match the natural numbers one to one with the lines just as before. The only change that is instigated by the addition of an extra line is the count starting from the new line and continuing from the previous beginning [Figure 2] (Vsauce, 2016).

Figure 2 (Vsauce, 2016)

Although it may be peculiar to grasp at first, the amount of lines hasn’t change. To prove this, I can add (n) additional lines and always end up with aleph-null lines. In the extreme case, I can even add another aleph-null amount of lines and still not affect the quantity as a line still exists for every natural number. We can showcase this phenomenon more clearly using a hotel and its clients. For our base case, lets consider a hotel with an infinite amount of rooms at full capacity. A new client shows up and requests a room; he is met with an emphatic yes and each occupant slides to the adjacent room. This results in the first occupant moving to the second room, the second occupant moving to the third room, and so on all the way up (Manchak, 2016). Essentially, the hotel has gone from completely occupied to having one free room that will be occupied by the new client. Through all of this, there must be some distinguishing feature between the collection of aleph-null lines and the aleph-null plus one case. We can work towards this contrast by numbering each line according to the order it was drawn in instead of matching the naturals one to one. Starting at zero, we number from left to right in ascending order until we “reach” the extra line. What number is assigned to the extra line? Labeling things in order is pretty different than counting them in the realm of infinity. Since the additional line doesn’t contribute to the total, we aren’t able to label it with a natural number. In order to label the line in the order it appeared in, we need a set of labels of numbers that extends past the naturals (Koch, 2014). We need ordinal numbers.

After exhausting the infinite collection of every possible counting number, we encounter the first transfinite ordinal: Omega (ω) (Koch, 2014). Rather than measuring magnitude, this ordinal number tells us how things are arranged. This arrangement, known as the order type, is vital to classifying a set. Deducing the order type of a set is quite simple: it is just the first ordinal number not needed to label every thing in the set in order. So for finite numbers, cardinality and order type are the same (Koch, 2014). The order type of all the naturals is omega and the order type of the following sequence is (ω + 1) and so on. No matter how long an arrangement becomes, as long as its well ordered, meaning every single part of it contains a beginning element, the whole thing describes a new ordinal number (Vsauce, 2016) [Figure 3]. This logic will be vital later on.

Figure 3 (Vsauce, 2016)

As we’ve explained the principles needed to comprehend logic outside of infinity, you might have questioned if we can really follow up about the endless process, natural number + 1, with something such as omega. Lucky for us, the things we assume to be true in mathematics aren’t more likely to be true if they better explain or predict what we observe (Vsauce, 2016). The things we assume to be true in mathematics are known as axioms. Essentially, axioms provide the flexibility needed to make assertions such as omega: we take a stand and say its true because we say it is; its consequences become what we observe. If the axioms we declare to be true lead us to contradictions or paradoxes, we can go back and tweak them or just abandon them all together (Vsauce, 2016). “In making sure of the axioms we accept don’t lead to problems, we’ve made math into something that is, unreasonably effective in the natural sciences.” (Vsauce, 2016). The axiom of infinity is just the declaration that one infinite set exists: the set of all finite numbers. If you don’t accept it, that’s fine, but if you accept it you can go pretty far (Vsauce, 2016).

Possessing this new tool-kit of information might lead you to believe that (omega + 1) is the biggest possible number. However, you should be careful as ordinal numbers just refer to an amount of stuff arranged differently. Omega + 1 isn’t bigger than omega; it just comes after omega. If aleph-null isn’t the end, then how can we show that there lies something beyond infinity? One of the best ways to do this is with Georg Cantor’s diagonal argument (Minutephysics, 2012). Let’s get to know the power set of aleph-null.

Meet the (in)-finities

The power set of any set S can be written as: P(S). This function represents all the different subsets you can make from set S. For example, from the set of two and three: I can make a set of nothing {}, or two{2}, or three{3}, or two and three {2,3}. As you can see, a power set contains many more members than the original set. Let’s be a bit ambitious now and ask, “What’s the power set of all natural numbers?” To answer this, let’s imagine a list of every natural number [Figure 4]. The numbers on the horizontal axis represent the various subsets; subset zero is the subset of all odd numbers while subset three is a subset of all numbers except five. The values in a subset are distinguished by a green (y) while those not in the subset are assigned a red (n). Obviously this list of subsets will be infinite but now imagine matching them all 1 to 1 with a natural number and producing a new subset that is clearly not listed anywhere. The result will be a set with more members than there are natural numbers: a bigger infinity than aleph-null.

Figure 4 (Vsauce, 2016)

Figure 5 (Vsauce, 2016)

The power set of the naturals, S, will always resist a one to one correspondence with the naturals. Not only is subset S infinitely bigger than aleph-null, repeated applications of power set with the previous power sets will produce power sets that can’t be put into one to one correspondence with the last. Not only is this a great way to produce bigger and bigger infinites, it means the that there are more cardinals after aleph–null! As we try to reach these distant cardinals let us remember that after omega, ordinals split and these numbers are no longer cardinals. They don’t refer to a greater amount than the last cardinal we reached. However maybe they can take us to one (Vsauce, 2016)
Beyond infinity

Once you get past cardinal numbers and through ordinal numbers, you eventually reach another ceiling that is omega plus omega (2ω) [Figure 6].

Figure 6 (Vsauce, 2016)

Going all the way to reach omega plus omega would mean creating another infinite set, but the axiom of infinity guarantees only one exists [Figure 6]. Does this mean we’ll have to add an axiom every time we describe aleph-null with more numbers (Vsauce, 2016)? No, the axiom schema of replacement can help us here (Vsauce, 2016). This assumption states that if you take a set, say aleph-null, and replace each element with something else, like shoes, what you’re left with is also a set (Vsauce, 2016). This is incredibly useful; lets now take every ordinal up to omega (denoted by area with a checkmark) and then put omega plus one. This replacement allows us to not only reach omega plus omega, but we can make jumps of any size we want so long that we only use numbers we have already achieved (Vsauce, 2016). We can replace every ordinal up to omega with omega times it to reach omega times omega. The axiom of replacement allows us to construct new ordinals without end. Eventually, we run out of standard mathematical notation as we continue this process but this of course isn’t the end.

Figure 7 (Vsauce, 2016)

Figure 8 (Vsauce, 2016)

We can still reach higher infinites using the replacement axiom; taking any ordinal we already reached, like say omega, we can jump from aleph to aleph and reach aleph omega [Figure 9] (Vsauce, 2016).

Figure 9 (Vsauce, 2016)

Figure 10 (Vsauce, 2016)

Figure 11 (Vsauce, 2016)

Figure 12 (Vsauce, 2016)

As we’ve attempted to journey past infinity, have we actually left the first infinity? Technically speaking, you can’t even reach aleph-null from below either. Since all numbers less than it are finite, any mathematical manipulation you perform on a number will only result in a finite amount. The only way we could access aleph-null is by declaring its existence axiomatically; the same thinking will be needed to access an inaccessible cardinal (Vsauce, 2016). The jump from nothing to the first infinity is parallel to the jump from infinity to an inaccessible cardinal (Vsauce, 2016). Numbers bigger than inaccessible only exist through a large cardinal axiom that asserts its existence. What’s amazing about these infinites is that they’re so big it is logical to question their existence as it is not clear if they can exist or be displayed in the physical universe (Vsauce, 2016) [Figure 12]. The question, “What is the largest number?” shifts to a more philosophical one which I think is great conclusion considering how most topics discussed in the course traced their roots to philosophy. Although we might never find the largest number, knowing that it may lie beyond our universe is a nice consolation.

Works Cited

Anderson, Kyle. "Aleph-0." -- from Wolfram MathWorld. N.p., 3 Feb. 2014. Web. 22 Apr. 2016.