Can One Infinity Be Greater Than Another?

What is infinity? The dictionary definition is a value which is unending or cannot be encapsulated by any known number. This means that nothing can be bigger right? Well there are mathematical proofs which show that some infinities are greater than others, the most famous being cantor s proof that there are more reals between 0 and 1 than all the natural numbers, integers and rationals.

To start off, let us use Hilbert s hotel as a thought experiment. Let us say we have an infinite hotel and every room has been occupied by a guest. A new guest arrives. Where can he go?

The solution is simple. We move every guest up by 1. Since there are an infinite number of rooms, there is no last room , so every guest should be able to find a new room. The same proposition can be made for if a guest leaves the hotel. Since there is an infinite number of guests, there is no last guest , so every room will still be occupied. This can be carried over to 2 new guests, 3 new guests, and any finite number of new guests.

Now let us bring an infinite number of new guests. Where can they go?

Since there is an infinite number of odd numbers and even numbers, we can simply move every existing guest to the room number that s twice their original, and that will leave all the odd numbers, where the infinite number of guests can go. This idea carries on with 2 infinite buses filled with guests. We move each guest currently in the hotel to the room 3 times their original, so we have 2 spaces between each guest. We then fill the spaces accordingly, moving the guests in the first bus to rooms 3n+1 and the guests in the second bus to 3n+2, where n is the seat number of each guest. Again this carries on to every finite multiple of infinity.

What will happen then if we have infinite buses each carrying infinite people?

There are many solutions to this problem, but I ll go over 2 of them. The first solution is based on the principle that we have an infinite number of primes 1*. We let all the people in the hotel move to room 2n where n is their corresponding room number. So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, and so on. The first bus sees each passenger move to the room 3n where n is the corresponding seat number. So the passenger in bus 1 seat 1 moves to room 3, the passenger in bus 1 seat 2 moves to room 9, and so on. This carries on with each bus number using its corresponding prime as a base.

This idea can be carried on to higher dimensions, so now let us say we have an infinite number of car parks with infinite buses carrying infinite people. The passenger in park 4 bus 5 seat 7 would go to room 7^11^7, as the 4th prime is 7 and the 5th prime is 11. lt;/p>While I believe this concept is very elegant and easy to get your head wrapped around, I believe it is not the most clean solution. You are still leaving many rooms empty, for example room 6, which in a way suggests that infinity >& infinityn. This in my opinion is flawed, because if we take the infinity2 example, we can technically fill up the hotel with only the first bus, and there would infinite buses left over. We can layer that logic over if we increase n. That is why I personally prefer the solution below.

(Note, I prefer this solution because it leaves no rooms empty which I believe is k

Figure 1. Fitting all the infinity2 within infinity.

This illustration goes through all passengers, and it doesn t allow any room numbers to be empty. This idea can be translated onto an infinite cube, an infinite 4 dimensional cube, etc.

Despite using these tricks, we have not yet found a way to map every passenger in the expression infinityinfinity into the hotel, which suggests that infinityinfinity is greater than just infinity. 2*

1* We use primes because they only have 2 factors (1 and itself). Therefore powers of primes only have factors of other powers of that particular prime. If we use powers of 4 for example, they will all already be covered by powers of 2.

2* A set is defined as being greater than another if we cannot match every member of that set with every member of the other set. We have not found a way to match every member in the set infinityinfinity with every member in the set of just infinity.

Arguably the most famous proof to show there are multiple levels of infinity is Cantor s proof that there are more real numbers between 0 and 1 than there are integers. The definition of a set being a different size to another is if we cannot match each term within the two sets one to one.

Here s the rundown of the proof:

Let us list all the natural numbers on one side (that is 1, 2, 3, 4, ) and random infinite strings on the other side. We ensure no infinite strings are repeated in the process. After all the natural numbers have been listed, we take the diagonal from the infinite strings to help us list out a new infinite string. The first number is different from the first number in the diagonal, the second number is different from the second number of the diagonal, etc. lt;/p>Below I have illustrated the

Figure 2: This shows that the natural numbers run out before the real numbers

This concept allows us to mathematically show that greater infinities do exist. The set of natural numbers essentially "run out" before the set of real numbers, therefore the set of real numbers is indeed greater than the set of natural numbers.

However, I am not too satisfied with this proof. While it is true that for a finite value, the new string will be different, we are working with an infinite value. This means the string has no final number, nor does the list of strings ever end. We can never list out all of the natural numbers, because the natural numbers, despite all being finite, are infinite in number. lt;/p>

While the set of real numbers is proved to be greater than the set of natural numbers, the set of rational numbers and the set of integers are proved the same size as the set of natural numbers. First, let us prove that the set of integers is the same size as the set of natural numbers. Since the set of integers is just the set of natural numbers and the set of negative integers and 0, we essentially just have the case of 2*infinity, which can be mapped in a one to one correspondence with 1*infinity. (See paragraph 1 on how).

What about the rational numbers?

A rational number is a fraction in its simplest form. We can list out every rational number by drawing an infinite square. As we go down the square, the denominator (bottom number) increases and as we go from left to right, the absolute value of the numerator (top number) increases. (0 itself is in a separate bubble, but it s just one term, and infinity+1 = infinity). Below is a diagram to illustrate

Figure 3: This diagram shows how we can correlate the rational numbers with infinity2 which we have already shown is the same size as infinity

What we are essentially dealing with here is infinity2, which as we have already seen can be mapped in a one to one correspondence with infinity. You might notice that there are a lot of fractions which are not in their simplest form. However, that doesn t matter because we have shown that the set of rational numbers even with extra terms is still the same size as the set of natural numbers. Moreover, we can just skip & over those terms when tracing a line through every term in the diagonal fashion.

So far, we have shown that the set of rational numbers is the same size as the set of integers and the set of natural numbers. However, the set of real numbers is greater than those sets. Here is the main logical flaw in this statement. Since infinity is a value which has no cap, theoretically nothing can have a greater cardinality. Therefore infinity must be the greatest value, and hence the set of rational numbers has the same size as the set of real numbers. lt;/p>Another problem I see with the claim is the double standard that people put when talking about whether one infinity is greater than another. The set of rational numbers is a subset of the set of real numbers, yet deemed smaller, and the set of natural numbers is a subset of the set of rational numbers, yet deemed the same size. While Cantor states that we can create a new infinite string that hasn t yet been listed after all the naturals have been listed hence proving the set of reals to be bigger than the set of naturals, what s to stop me taking all the natural numbers from the set of rationals, place them in a one to one correspondence with the set of natural numbers, and still have an infinite amount of rationals left.

I believe that a key flaw in Cantor s proof that the set of reals is greater than the set of naturals is that the natural numbers will run out before the real numbers. This of course makes sense, due to the sheer amount of possible real numbers compared to the natural numbers. However how can something infinite ever run out? I think that while Cantor s proof is mathematically correct and valid, it is trying to prove something which we as humans never can. Infinity does not dwell among us, and we can never truly know the nature of it with our finite minds. Trying to prove that there are greater infinities in my opinion is like trying to prove God whether exists or not. You can only believe whether there are greater infinities or not. We can never truly know.

In my opinion, there is only one infinity. I believe that it makes no logical sense that there are infinities greater than the one which we normally think of when we say infinity . Infinity is never ending, and therefore I believe we can match any other infinite set in a one to one correspondence with it, since both sets can never run out of terms. While most mathematicians disagree, and think that we cannot match the reals and the integers in a one to one correspondence, in the end, we as finite beings can never truly know the nature of infinity.

BIBLIOGRAPHY

https://plus.maths.org/content/hilberts-hotel - Marianne Freiberger, Feb 13 2017