# Is An Actual Infinite Coherent? Part 1

Post Author:  Darrell

In short, no.   The story of Hilbert’s Hotel helps to demonstrate this fact.  It goes like this…  Let’s say we have a hotel that has an infinite number of rooms and an infinite number of guests; as a result, the hotel is full.  If a prospective guest walks in and asks for a room, can he check in?  Since there are an infinite number of rooms, the answer must be “Yes.”  How about if an infinite number of guests arrive wanting to check in.  Can they?  Again, despite the fact that the hotel already has an infinite number of guests, since there are an infinite number of rooms, guests can always check in – even an infinite number more.

Now, let’s say that the guests in all of the odd-numbered rooms check out, how many guests are left?  There are an infinite number of total rooms.  However, there are also an infinite number of odd-numbered rooms, the guests of which checked out, and there are also an infinite number of even-numbered rooms, the ones still left occupied.  So in reality, there are still an infinite number of guests left in the hotel even though an infinite number of guests just checked out.  This means when you take an infinite away from an infinite, you still get an infinite.

Where does this leave us?  Even though Hilbert’s Hotel has an infinite number of guests and rooms, more rooms and guests can always be added.  In addition, no matter how many guests check out there will always be an infinite number of guests left.  As a result, the hotel could have a sign which reads, “Hilbert’s Hotel: Always full, Yet Rooms Are Always Available.”

This illustration points out how an actual infinite is incoherent.  In an actual infinite the whole and the parts are always equal.  You can take half away and still have an infinite, or you can add more and still have the same amount – an infinite.  However, in reality, a part can never equal a whole.  For example, two is part of four (half of it to be exact). Thus, two can never equal four.

Is there any infinite that is coherent?  Yes, a potential infinite.   A potential infinite is always finite and the whole is always greater than the parts.  In a potential infinite you can always add more, but it will never become actually infinite.  For example, let’s say you have 100 Jelly Beans in a pile.  You can always add more Jelly Beans to the pile.  In fact, you can continue to add Jelly Beans and never reach a maximum.  As a result, you could say that the pile you are creating as you add more is potentially infinite.  It is not an actual infinite because no matter how many you add, there are always a finite number of Jelly Beans in the pile.  However, it is potentially infinite because more can always be added.

In the next post, we will look at how the concept of an actual infinite applies to the universe, time, creation, and Mormonism.  Stick around.

Darrell

• davidgawthorne

Hi, Darrell,

I was just passing and thought I would stick my nose in.

When you say that an actual infinite is incoherent, do you mean that it is contradictory? I do not see a contradiction anywhere in the example of Hilbert’s Hotel, just a very weird scenario. However, maybe infinity just is weird and should not be compared to the behaviour of finite values by way of reductio. How would you respond to this?

• Hello David,

Thanks for stopping by. To answer your question, yes I do mean that it is contradictory. In an actual infinite the whole and the part are equal, e.g., half of an infinite is still an infinite. In the Hilbert’s Hotel example this was pointed out by the fact that the hotel would be both full and still have vacancy… a both/and contradiction.

Take Care!!

Darrell

• David Cobb

Please correct me if I am wrong, but I seem to remember from my graduate mathematics studies, that “infinties” have degrees, that is, some “infinties” are larger than others. Consider for example, the infinite set of integers, and compare to the infinite set of rational numbers, by necessity, the infinite set of rational numbers is much larger than the infinite set of integers, and thus a 1-1 correspondence could not exist with respect to integers and rational numbers.

• davidgawthorne

Sorry, how is that a contradiction. What proposition comes out both true and false?

• The Hotel is both full and not full.

Darrell

• David, I was hoping someone who knows some mathematics would weigh in. I get your point about “degrees of infinity” eg “the set of all integers compared to the set of all even numbers. The set of all positive numbers + the set of all negative numbers = the set of all possible numbers (every term infinite, and yet decidedly non-equal)
I think this distinction is the one that gives us the necessity of an absolute God being one and only one. If we posit Him as a “fundamental infinity” then there can be only one. There can be many infinite lines, many infinite planes, but only one infinite volume (yea, let’s leave out higher-dimensional physics please!)

I’m not sure where Darrell is going, and am VERY eager to read. But at great risk, I might start with the idea that infinity, like God himself, is coherent as an axiom. If it seems incoherent, see rule #1. There is some aspect I haven’t yet got my arms around. For instance, in Hilbert’s Hotel, perhaps “infinite” means something like “all rooms possible”, rather than an undefined point on a number line. Then (unless we bump into an absurdity, like the heavy rock question), infinity becomes something like all that are needed, and all that can possibly be needed. An infinite number becomes something other than just a number, but relates in some way to something else.
I expect there is a more mathematically rigorous way to express what I mean, but I suspect it reveals something more of the character of God, His infinite love and provision.

I await with baited breath episode 2!
R. Eric Sawyer

• David,

If you take all the positive numbers (2, 4, 6, 8, etc.) and compare them to all natural numbers (1, 2, 3, 4, etc.), you still have a 1 to 1 coorespondence if you take it to infinity. 1 cooresponds to 2, 2 cooresponds to 4, 3 cooresponds to 6, etc. Since there is never an end, you have one to one coorespondence.

One to one coorespondence is one of the properties of an actual infinite. As a result, if you have a set that does not have a one to one coorespondence with all natural numbers, you actually don’t have an actual infinite. You may in fact then be talking about a potential infinite… which is finite. You could have two potential infinites where one is greater than the other. You would know this by comparing their finite amounts.

Darrell

• davidgawthorne

But that is not correct. The hotel is full at all times.

• Can someone check in?

Darrell

• davidgawthorne

I’m not sure whether or not this was intended for me, but I’ll bite.

Yes, someone can check in.

Be careful not to equivocate between two available senses of ‘full’. If ‘full’ is defined as something like ‘not being capable of taking more guests’ then Hilbert’s Hotel is never full. If ‘full’ is defined as something like ‘having each adjecent unit space occupied’ then Hilbert’s Hotel is always full unless you substract the right infinite set of guests. So, the Hotel is both full and not full, but only if we are using different senses and of ‘full’, so, there is no genuine contradiction.

• Tye

The hotel is full in and only that there are no empty rooms. However, since there are an (actual) infinity of rooms you may add more guests without creating more rooms and yet still have an infinite number of guests. All of the rooms in Hilbert’s hotel are full and yet guests can be added to rooms without doubling up. This is the contradiction. Also, as it has already been pointed out, infinite subtracted from infinite is still infinite. There are some things that make sense in mathematics such as the concept of a potential infinite (and imaginary numbers to give another example) that do not correspond to reality. You cannot have the square root of negative one apples. And despite what hawking claims in his model for the universe, there is no “imaginary time”.

Consider Hilbert’s hotel again. All infinite rooms are occupied. All guests in odd rooms leave. Exactly one half of the guests have now left. However, the number of guests in the hotel is exactly the same as it was before. It gets better. If we shift every guest down rooms (ie 2->1 and 4->2 etc) then all of the rooms are now full again. This is the absurdity of Hilbert’s hotel.

Another example. Say two planets, Planet A and Planet B orbit a sun. It takes planet A one Earth year to complete its orbit and Planet B 10 earth years. After infinity time, Planet A has orbited the sun 10 times more than Planet B. Yet at the same time, they have both completed the same number of orbits. In fact, they have ALWAYS completed the same number of orbits since they have been orbiting from infinity, they have always completed infinity orbits. The best part is this- when asked whether the number of orbits is even or odd, mathematics dictates that it can be neither. It is both. The Planets have completed a number of orbits that is mathematically even and odd at the same time.

• David Cobb

First of all, let us not draw an “equivlanec relation” between the idea of infinity in mathematics, and the philosophical idea of the infinity of God, these two are apples and oranges. As to “degrees of infinity” mathematically, I have absolutely no doubt. Again, consider any two consecutive integers on the number line, there exists between those two integers in infinite number of rational numbers. (Assuming everyone knows the definition of a rational number.) QED. My only trepidation in this matter is whether or not this theorectial axiom of “infinities” applies in this particular circumstance. I am familiar with Hilberts work in geometry, which led to implications and advancements in theories of non-euclidean geomtries, but I am not familiar with “Hilberts Hotel”. The the most prolific mathematician in Set Theory was Kurt Godel. Many contributions to the mathematical notions of infinities.
Also keep in mind, all mathematics, even at the supposed “concrete” level of numerals and arithmetic, in reality is completely abstract. Example, if I asked you to “show me” the number 4, the likely response is to demonstrate 4 fingers etc. However, that is not “4”. “4” is an abstraction, a creation of the mind of man. (Granted God is the one who granted the gift of intellect).

• David Cobb

Tye

“Imaginary” values are indeed quite real. Simply mathematical terminology to indicate a different “set” of numbers. “Imaginary” values are routinely encountered in electrical engineering, if fact “imaginary” current will kill really dead, really fast. “Imaginary” is simple a “term”.

• David Cobb

In fact, the “real” numbers are a subset of “imaginary” (complex numbers). Complex numbers are of the form a+bi, where a, b are real values and i = (-1)^1/2

When a=0, then bi is a purely imaginary value, when b=0, then you have a typical real number.

• davidgawthorne

You still have not stated which proposition and its negation constitute your contradiction.

Apparent absurdity does not a contradiction make.

I do not grant you the premise that Planet A has orbited the sun 10 times more than Planet B after an infinite period. If infinity is a number then that is the number of times that both planets have orbitted and that is the same number.

In relation to the question of whether the orbits are odd or even, this has a trick to it. One cannot start from a finite number and by successive addition of finite numbers acheive infinity. For, the addition of no two finite numbers results in infinity. Thus, the planets would already need to have been orbiting for an infinite period of time for their them to orbit an infinite number of times. So, whether infinity is odd or even itself determines whether the planets have orbitted for an odd or an even number of times.

• David,

If “every adjacent unit space is occupied” then the hotel by nature does not have any adjacent space available for someone to check in. Thus, the hotel is “not capable of taking in more guests”. However, due to the fact that there are an infinite number of rooms, it is also not full because the hotel can still take in more guests. Thus, the hotel is both full and not full.

There are other contradictions that come into play with an actual infinite as well. For example, half of an infinite is both half of an infinite and equal to an infinite.

Darrell

• First of all, let us not draw an “equivlanec relation” between the idea of infinity in mathematics, and the philosophical idea of the infinity of God, these two are apples and oranges.

Absolutely. One is speaking of an actual infinite number of things while the other is speaking of a metaphysical infinite. These are completely different things.

Darrell

• davidgawthorne

The hotel does not have any space until every current guest shifts to the next room, then it is capable of taking more guests. This might be weird but it does not present a contradiction.

It is not a contradiction to say that an infinite is both half of an infinite and equal to an infinite, just as it is not a contradication to say that zero is both half of zero and equal to zero. Half of infinity just is infinity.

We seem to be going in circles, so I might just quit here. Thanks for taking the time with me, though.

• The hotel does not have any space until every current guest shifts to the next room, then it is capable of taking more guests. This might be weird but it does not present a contradiction.

Shifting down to the next room necessites an empty room to be shifted into. So it implies that the hotel is both full and not full.

It is not a contradiction to say that an infinite is both half of an infinite and equal to an infinite, just as it is not a contradication to say that zero is both half of zero and equal to zero. Half of infinity just is infinity.

I don’t really think this follows as there is no such thing as half of zero. You can’t split “nothing” in two as there is “nothing” to be split.

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Thanks so much for the conversation. It is certainly enjoyable!!

Darrell

• David Cobb

Really looking forward to “wherever” you are going with this, particularly with respect to time.

• David Cobb

The idea that infinite sets might not be equal, but instead may be ordered in terms of degrees is one that seems overtly counterintuitive. After all, how can one infinite number possibly be greater than another infinite number?
The counter-logic to this, though, is incredibly intuitive, and deceptively simple. First, some terminology should be cleared up. With the idea of “degrees” of infinites, a set of terms arose in order to define them – the aleph numbers.

The first degree of infinites is known as aleph-null (from the first letter in the Hebrew alphabet). The second is aleph-one and so on.

The fundamental ideas here are those of “countability” and “one-to-one relationships”. The standard, aleph-null infinities are those which are considered “countably infinite,” which may very well seem like a logically contradictory term, but is not so.

Countable Infinites
Countable infinity simply means that the sequence of infinite numbers can be counted sequentially (though without end). The integers, then, are aleph-null numbers, as they can be counted: 1,2,3,4,5,6,n+1…

The even numbers: 2,4,6,8,10,n+1…

The primes: 1,2,3,5,7,n+1…

These numbers can all be counted without end, and they all succomb to a one to one ratio with each other – that is, the elements of each set can correspond to elements in the others, and progress in an orderly fashion forever. For instance, the one to one ratio of integers and even numbers begins: 1 to 2, 2 to 4, 3 to 6, 4 to 8. The first integer is matched with the first even number, the second with the second, and so on.

While the numbers themselves may not be equal, this system can continue forever, the ratio remaining the same.

Uncountable Aleph-One’s
What, though, of those infinity of numbers in between two integers, such as 0 and 1? After all, with just a bit of rational thought, it becomes clear that in between these two numbers possesses an infinite set of other numbers (0.01, 0.114, 0.00000456, 0.32432423423344, etc…).

While these numbers are also infinite, they are considered by mathematicians to possess a higher degree of infinity (aleph-one), as they are uncountable (because in between any two of them, there is always an infinite number of others), and cannot be counted one to one with any infinite set.

Can one possibly say, then, that the number of aleph-one numbers are greater than aleph-null numbers? Not exactly. They are both infinity. One can be considered greater than the other, however, and that is what makes this idea of cardinality of infinite sets so mathematically interesting.

The cardinality of infinities does noe end here, of course. This is just one example of a conclusion that might be drawn from this idea. It should be able, at the very least, to allow one to realize that perhaps not all infinities are equal, and that mathematicians have their work cut out for them in making sense of these things.

Read more at Suite101: Degrees of Infinity: The Inequality of Endless Sets http://mathchaostheory.suite101.com/article.cfm/degrees_of_infinity#ixzz0f64geoQW

Note the “nod” to God in the domain of mathematics, that being, use of the hebrew alphabet for articulation of infinities. (aleph…._

• Boz

During my university mathematics courses, we did some work on infinity. This is some of what I remember.

We will call ‘inf’, all of the positive integers. The way to discover if a particular description of infinity is equal to ‘inf’, is to map the positive integers to the new description. If a mapping is possible, then the new description is equal to inf. The hotel example is an easier way to understand mapping.

e.g. inf x 2 = inf. 1 maps to 2, 2 maps to 4, 3 maps to 6, etc.

inf x n = inf
inf + n = inf
inf + inf = inf
inf / n = inf
inf / inf = undefined
inf / 0 = undefined
inf*inf = inf
inf^n = inf

I also remember that there are several different sizes of infinity. That means that there is a description of a number (A), and a description of another number (B), and both are infinite, and B has a larger size than A

An example of a larger infinity is that of all numbers that can be expressed as a decimal. These are called Real Numbers. The infinity of real numbers is larger than the infinity of positive integers (inf).

So, if one person arrived at the hotel on behalf of every real number in existence, they would not be able to fit in the hotel.

Darrell, what is your definition of full in this story, such that a incoherency exists? It looks like you might be using equivocation.

• Boz

David Cobb said: “While these numbers are also infinite, they are considered by mathematicians to possess a higher degree of infinity (aleph-one), as they are uncountable (because in between any two of them, there is always an infinite number of others), and cannot be counted one to one with any infinite set.”

The set of rational numbers (fractions comprised of integers) is countable, (i.e. equal to aleph null), yet in between any two of them, there is an infinite number of others. Your definition of uncountable is incomplete.

• Boz,

My definition of full is the standard definition… every room has someone in it. Where does the incoherency come from… from the fact that people can still check in because rooms are still available.

The hotel is both full and not full. There is no equivocation.

Darrell

• With all of the discussion about mathmatical infinites, I felt impressed to point out that there is a difference between the concept of an infinite, e.g., mathmatical concept, verses an actual infinite.

Concepts are entirely cognitive, and in many cases, they have no basis in reality. As a result, the concept of an infinite number of odd numbers verses an infinite number of all real numbers or integers really has no basis in reality. Show me an actual odd number. Show me a real number. Show me an actual integer. What are they? You can’t do it because they are formed wholly in your mind. They are concepts formed in our mind to help us understand, conceive, and explain reality, but they are not real things in and of themselves.

In contrast, when we are talking about an actual infinite number of things, we are talking about a real thing, e.g., Jelly Beans.

———————————-

In addition, there is another way of explaining how an actual infinite cannot be real: an actual infinite can never be reached, for you can always add one more. On the other hand, a Potential Infinite is possible. But keep in mind a Potential Infinite is always finite.

Darrell

• Boz

Darrell,

when you define full as (1)(every room occupied) – the hotel is [full], and the hotel is not [not full].
when you define full as (2)(people can check in) – the hotel is [not full], and the hotel is not [full].

this is equivocation because you are switching the definition of a word part way through your argument.

here is another example:

“all banks are adjacent to rivers” is both true and false.

By definition, a bank is adjacent to a river. But when I go to deposit my money into my bank account, I am nowhere near a river.

Maybe an actual bank is incoherent, and cannot possibly exist? :p

• Boz,

It is not an equivocation because I am not changing the defintion of the word full. Instead, I am using what the definition of full means, and it leads us to a contradiction. For, if every room is occupied there are by defintion no rooms available for people to check into.

———————————-

By defintion: all rooms occupied = no rooms available

—but in this scenario it means the above plus —

all rooms occupied = rooms available

———————————————————
Therefore, since the hotel is full, but people can still check in, it is incoherent. Perhaps even a better way to word it would be, “The hotel has all rooms occupied, but there are still rooms available for people to check in.” This is a both/and contradiction.

Darrell

• Boz

darrell, I’m trying to get across your new argument.

would it be fair to summarise your argument as:

“(1)In this scenario, if all rooms are occupied, no rooms are available.
(2)In this scenario, if all rooms are occupied, some rooms are available.”

• I think you understand what I am saying.

All rooms being occupied by nature means that there are no rooms to check into. However, the fact that someone can check in by nature means there are rooms available to check into. As a result, the claim that all rooms are occupied is contradictory to the fact that someone can check in.

Darrell

• Boz

It is false to claim that, under this scenario, “All rooms being occupied means that there are no rooms to check into”

Darrell

• Boz

your claim was that, under this scenario, “All rooms being occupied means that there are no rooms to check into”

if all rooms are occupied, you can move everyone up one room (room 1 -> room 2, etc), which gives you one room to check in to.

so, if all rooms are occupied, and a new guest arrives, there is a room for them to check in to.

• If all rooms are occupied, by nature there is no vacant room to be shifted into. However, in order for the shifting to create a vacant room there must be at least one vacant room for one of the guests to shift into. Otherwise, the shifting would simply be among the rooms in which the guests occupy, i.e., the guests would simply switch rooms, and you would thereby still be left with all rooms being occupied. Shifting cannot CREATE a vacant room. There must be a vacant room BEFORE the shifting takes place.

Darrell

• Boz

When considering the claim – “under this scenario, All rooms being occupied means that there are no rooms to check into” , It doesn’t matter how the extra rooms are achieved.

It is a fact that, under this scenario, extra guests can be accomodated by shifting. [you have agreed with this earlier]

which shows your claim to be false.

I suspect our disagreement may come down to your claim that “Shifting cannot CREATE a vacant room.”

Create is an inappropriate word, because the hotel does not change size, there are always ‘inf’ rooms. No rooms are created or destroyed.

shifting can remove guests while all rooms remain full (move everyone down 10 rooms)
shifting can achieve extra vacant rooms for new guests (move everyone up 10 rooms)
shifting can achieve no entries or exits. (everyone switch with their neighbour)

I hope i’m bveing clear here

• It is a fact that, under this scenario, extra guests can be accomodated by shifting. [you have agreed with this earlier]

Not exactly. Shifting cannot accomodate more guests unless it somehow creates more rooms as all the rooms that exist in the hotel are currently occupied by the infinite number of guests currently in the hotel. Thus, there are no rooms to be shifted INTO as all are occupied. The only way to accomodate another guest is to add another room.

I have heard the explanation you are offering before. Basically, it is just a fancy way of saying you are adding another room. However, the fact that another room can be added goes right to my point about how an actual infinite number of things is nothing more than a concept. It is not a reality. In reality, all infinites are potential infinites. They are never actual because you can always add one more.

Darrell

• Hi, interesting post.

In mathematics we use the concept of infinity, but the concept you call potential infinity. When quantities increase and increase without a finite limit, we say that they tend to infinity, or that their limit is infinity. Others, however, prefer to say that the quantity does not have a limit, because they say infinity is not a real concept.

Just passing by, just wanted to say that I like your perspective on this one.

Pablo

• Thanks Pablo!

Have a great day!

Darrell