Monday, July 28, 2008

Can an Actual Infinity Exist?

Can an actual infinity exist?

I have been recently reading William Lane Craig at his website, Reasonable Faith. I also have the second edition of his book, Reasonable Faith, but he is coming out with his third edition of the book, which significantly expands the material.

One of Dr. Craig's specialties is the Kalam argument for the existence of God. His website, under the scholarly section, has several articles discussing the Kalam argument, along with discussing critiques people have made about the argument. The article I am reading is here.

One of the central arguments in the Kalam argument is that there cannot be an actual infinity. Dr. Craig cites an example of the weirdness of infinity by discussing the Hilbert Hotel.

Imagine a hotel that has an infinite number of rooms. The first room is numbered 1, the second is numbered 2, the third 3, etc. You are a traveler and you come to the hotel and ask for a room. The manager at the desk informs you that all the rooms are occupied, the hotel is full. However, the manager is clever, he says he will make a room available for you. He will have the person in room 1 move to room 2, the person in room 2 move to room three, etc. Thus, without adding any rooms or removing any occupants, the hotel that was once full now has a room.

But it gets stranger. The manager has a sudden insight. I will move each person in each room to 2 times the room number. So the person in room 1 moves to room 2, the person in room 2 moves to room 4, the person in room 3 moves to room 6, the person in room 4 moves to room 8, and so on. Now without adding any rooms or removing any persons, the hotel that was full now has an infinite number of empty rooms -- as well as having an infinite number of occupied rooms.

It is William Craig's contention that such an infinity cannot exist in reality. Part of Dr. Craig's contention is that we have the contradiction of two equal numbers, infinity, being in a contradictory state of where one infinity becomes less to the same identical infinity.

My critique of Dr. Craig's comments is that mathematicians do not speak of an infinite number which can be compared to other infinite numbers in the same way as finite numbers are compared. Otherwise the branches of mathematics which work with infinite sets would have logical inconsistencies in them -- and mathematicians hate logical inconsistencies in their mathematical systems and study hard to remove them.

But Dr. Craig does bring up a point which I have not thought much about. Can an actual infinity exist in the real realm of the universe? I think I am beginning to see his point, that it cannot. But I want to think why it cannot. I have some vague notions and I will sleep on it and continue the discussion later.



18 comments:

Celestial Fundie said...

I don't think an actual infinity can exist.

But one question in my mind is whether God's knowledge of the future would count as an actual infinity.

jared said...

The problem lies in the human incapacity to understand the concept and category of "infinite". In fact, that incapacity is the only reason there are "larger" infinite sets within set theory; it has nothing to do with Cantor, imo.

Earl said...

Matthew,

That is an interesting question to contemplate. Does God's knowledge of the future counts as an actual infinity. Assuming God has full omniscience of the future, it can be approached in two ways:

(1) The future is infinite, therefore God's knowledge of it must be infinite.

(2) The future is a potential infinite that is never achieved. As we go on in time, each moment can be examined to state a measure of time in the future. Every measure will be finite -- even though the numbers grow large.

Approach (2) in particular assumes that between any two different points in time has a finite set of sub-time elements in it. For our universe, it can be argued that there is a granularity of time which cannot be further subdivided. These are "Plank" units of time. They are very small, but theoretically they cannot be further subdivided. This would eliminate Zeno's Paradox -- but more importantly make the individual "time points" between any two periods of time finite -- thus allowing for approach 2 to hold.

So, I think it is entirely possible that God's knowledge of the future is not an actual infinite -- at least I think so these past 5 minutes. I might think differently 10 minutes from now.

Earl said...

Jared contra mundi -- or at least contra thousands of mathematicians :o)

However, you've got Leopold Kronecker (1823-1891) on your side, who founded the finitist school of mathematics which later developed into Intuitionism.

There are some significant issues with infinite mathematics -- such as Godel's theorems of Incompleteness.

However, I think that various forms of infinite and transfinite mathematics have been constructed that are consistent -- but certainly not complete. These live in the realm of abstraction and their application to the real universe is questionable at best.

But there does not deter many astrophysicists, such as Max Tegmark of MIT. He seems to view mathematics as real and the universe as a sort of window dressing to mathematics. Very Platonic, it seems to me.

Earl said...

...or maybe Platonic is not the word, rather Platonist, as in blatant Platonist.

jared said...

earl,

Being a mathematician, of course, you would know better than I. The problem I have is with the definition of "infinite" within mathematics. In mathematics infinite is, ironically, simply another type of limit; a way of measuring or relating or ordering. As a limit, it makes sense to me that some infinite set could be larger than (i.e. able to measure, order or relate to more than) some other infinite set. It is because mathematics works with such a definition of "infinite" that theorems like Cantor's and Gödel's can be logically derived and proved. And so my beef is not with the proofs, rather, it is with understanding the concept/category of infinite as a limit.

I like using Cantor's diagonal argument as an example. Cantor demonstrates that one is able to generate a number from within an infinite set that cannot be put in a one-to-one correspondence with the set of natural numbers. Any such set is, then, said to be larger than the set of natural numbers. Since the set of natural numbers is infinite, and this other set can include more numbers than the set of natural numbers it, then, is larger than the set of natural numbers. Hence, the infinite set of natural numbers is smaller than the infinite set of, say, real numbers because from the set of real numbers one can generate (using the diagonal argument) a number which has no natural number correspondent. The problem I have with this is because it seems no one has asked the following question: why can not one more natural number be generated to account for this newly generated real number? Supposedly it is because we have already used up all the natural numbers before generating this new real number. Since it is impossible to use up an infinity, it seems to me that there is no reason at all that this newly generated real number can't also have a correspondingly generated natural number.

I've always envisioned Cantor's internal discussion thusly:

"If I put set of all natural numbers into a 1:1 correspondence with the set of all real numbers first and then I create this new real number out of the set of real numbers that, then, must mean there are no more natural numbers to account for this new real number. So that must mean the set of real numbers is larger than the set of natural numbers!"

The problems glare brightly to me. Number one, you can't actually put two infinite sets in a 1:1 correspondence because of time. Number two, if the set of real numbers does not initially contain the newly generated number then the newly generated number can't be used as proof that the set of real numbers is larger than the set of natural numbers. This, in effect, is like cheating. It's like holding a real number in your pocket while you wait for the correspondence task to be done (which takes us back to the first problem) and then revealing this number after the fact and declaring victory. Number three, if the newly generated number is to be included in the set of real numbers, then it is also already in a 1:1 correspondence with the set of natural numbers via the initial task of of putting them in such a relation.

Now Cantor wants to say that the reals and the naturals can't be put in a 1:1 correspondence because one is able to generate "new" real numbers supposedly after the reals and the naturals have been arranged in such a correspondence. The problem here is that for every newly generated real number, there are (once again) an infinite number of natural numbers to attend to them. In other words, the problem is in assuming that an infinite set can have a cardinality, hence "infinite" being ironically (and miraculously!) changed into a type of limit. It doesn't help that infinite sets are poorly defined as any set which can be put in a 1:1 correspondence with a proper subset of itself (Wiki defines it more simply as any set which isn't finite).

I suppose this is a very long-winded way of saying that I think there are no outstanding reasons to accept the cardinality of infinity; or that I think there are no outstanding reasons to accept the concept of infinity as a limit. This, I believe, is the breaking point where infinite actually becomes self-contradictory as a concept; i.e. infinite becomes a limit (e.g. countable). In short (and again) this is what I meant by my first comment about our incapacity to understand the concept and category of infinite. Once you put infinity in a box, it does all sorts of fun and fascinating things: like not being infinity any more. But what do I know, not being a mathematician and all.

Celestial Fundie said...

But if God's knowledge of the future currently exists in God's mind it must surely be an actual rather than a potential infinite, surely?

God currently knows all future occurences.

Earl said...

Matthew,

Up until yesterday, I would fully agree with you. But I am beginning to see what Duns Scotus (1266-1308) was talking about, an unbounded knowledge that does not actually achieve infinity.

You are probably right, and in a few days I will return to sanity. I always thought the middle ages philosophers where "dunces" (Scotus being one of them) -- but I am not so quick now to dismiss them.

Earl said...

I cannot believe the major typo of my previous answer...

Jared,

Cantor's argument hinges in part on the definition of rational numbers -- a number that is represented by a quotient of integers. For any Rational number r, there exists an integer a and an integer b such that:

r = a/b

Cantor showed there is a way to construct a number, i, such that i cannot be represented by two integers. Because the set of Reals contains both Rational numbers (the set of numbers represented by the quotient of two integers) and Irrational numbers (those that can't), Cantor demonstrated that the set of Reals contains more elements that the set of Rationals. Thus there is no actual contradiction with these two types of sets.

In this context, more has a precise definition. A set Z is said the have more than set Y if Y is a proper subset of Z. That is, for every element y in Y it also is in Z, but there exists elements in Z that do not exist in Y. This is the case for Rationals versus Reals. The Real number set contains all the elements of the Ration numbers, plus more.

If we were to add the constructed number of Cantor's diagonalization to the set of Rational Numbers, that set would no longer be the set of Rational numbers, because that set now contains a number that cannot be represented as a quotient of two integers.


Jared, I am finally hearing what you are saying about infinites -- that there is something fundamentally fishy about them in the real universe.

Celestial Fundie said...

Earl, I hope you can explain this one to me, as this has troubled me since I was about 19 or 20.

Earl said...

Matthew,

I cannot explain this to my satisfaction yet. The ideas are so new to me. In the last few days I remembered some of my math classes where infinity was discussed and was avoided in the formal development of the area of mathematics.

From my hazy memory, infinity was approached as a limit condition but not actually reached in these models. For instance, in Calculus you can discuss a limit for a function as it approaches, say zero, and the function is undefined for zero. This allowed for defining derivatives for the function at zero even though the function was undefined at zero.

I am going to have to think about this more to come up with some reasonable explanations without extreme technical details.

So, I'll get back to you...

jared said...

earl,

Again, I apologize for my lack of mathematical prowess, but onward I push! The concept of "more" is unintelligible within the context of infinity. If something is infinite it can't have more; more is a meaningless descriptor. So even though the set of reals contains elements that are not included in the set of rationals and even though the set of reals includes the set of rationals, it does not stand to reason that the set of real numbers has more elements than the set of rationals because both sets are infinite. It will always, or never (depending on how you look at it), be possible to put the two sets in a 1:1 correspondence because they both have no end, i.e. they are both infinite. The only resort to solving such a problem is to redefine the concept of infinite (as Cantor does) so that it is more precise; or, rather, so that it is more limiting.


earl and celestial fundie,

I would submit that God's knowledge of the future is not an actual infinity for at least two reasons. First, God's omniscience is complete (nothing can be added and nothing can be taken away) whereas an infinity, by definition, cannot be (you can add to and take away from an infinity, though it doesn't change). Secondly, infinities have a beginning whereas God's knowledge does not (it being eternal as He is eternal).

Earl said...

Jared,

No apologies needed. Its one of the great things about you is that you push on. In doing that, you get more information.

You're right about the concept of more and less with infinite sets. It is better to speak of subsets and cardinality of sets which speaks more precisely about concepts that don't exactly match our regular words of more and less.

About 1:1 correspondence. The interesting thing about the set of Rational numbers and the set of Integers is that they can be put into a 1:1 correspondence. But, the set of Rational numbers and Real numbers cannot be put into 1:1 correspondence. Cantor proved they cannot, that is what the whole diagonalization proof was about. It is an astonishing beautiful and elegant proof. Part of the beauty and strikingness of this is the counter intuitive nature of all this -- that somehow these two different infinite sets cannot be mapped 1:1 at all.

This is what I have come to love about Mathematics -- being able to work with logically consistent things that are outside of our experience. Think of 4-dimensions, 5 dimensions, or higher. These are things we cannot imagine in our heads, but we can develop mathematics that describes such things. Further, it seems that General Relativity implies we that we live in a three-dimensional space that is curved in 4-dimensions or higher. How do you see if such a thing is real? With mathematics, you can make predictions about what space will look like and look for those things.

Your last comments are interesting. I'll comment on them a little later.

Earl said...

Jared,

Good analogical reasoning. All analogies have their points of comparison and where they break down.

God's omniscience is complete (nothing can be added and nothing can be taken away) whereas an infinity, by definition, cannot be (you can add to and take away from an infinity, though it doesn't change).

The point where it works is that concept of infinity as being a boundless limit is preserved in a set where you take away finite and even infinite numbers of elements from it. Thus an infinite set can be diminished without taking away the property of infinity, which is a measure of its numerabilty. This is not true of finite sets. It's sense of numerability will diminish as a result.

There is a sense of incompleteness in an infinite set if you take away an element. Take 3 out of the set of integers and there is a sense in which the set of integers is incomplete -- yet the infinity aspect of it is not diminished.

That said, I think that is a powerful point, considering the numerability issues.

infinities have a beginning whereas God's knowledge does not (it being eternal as He is eternal).

The set of natural numbers, {1, 2, 3, ...} has a beginning, but the set of integers does not {..., -3, -2, -1, 0, 1, 2, 3, ...}.

But I think you're on to something. After reading some more philosophy, I am gaining a greater respect for how philosophical analysis provides sharp insight into these issues.

Jared said...

earl,

Does not the set of integers begin with zero and then extend (infinitely) in both directions from there?

As an aside (though not completely irrelevant), it has always bothered me that mathematics cannot be content with the incomprehensibility of nothingness and infiniteness. In mathematics nothingness is simply another kind of something and infinite is simply another kind of limit; both concepts which dramatically alter their respective semantic content. For example, what's the difference between saying I have zero dollars and saying I have no dollars? From a mathematical point of view it can be said of the former that I have, in some sense, dollars which total zero whereas of the former I have nothing, i.e. I don't have dollars at all in any sense. Of course, from a practical standpoint they are the same, if I have zero dollars then I have no dollars. But in mathematics zero isn't nothing, just as infinite isn't unlimited. Does that make sense?

Earl said...

Jared,

Does not the set of integers begin with zero and then extend (infinitely) in both directions from there?

I see your point. Technically the set of Integers does not start at zero. But there is a "reflection" aspect of the set that pivots around zero.

Mathematicians like to play with these incomprehensible ideas. While we cannot comprehend each of these things, the surprising thing is that there are things that can be stated about them -- a limited set of properties that applies to each of these things.

I liked what someone gave for the definition of nothing: what a rock thinks.

Anonymous said...

Does God know all the real numbers? If so then his knowledge constitutes an actual infinity. If not, then he doesnt know everything (and actually only knows an infinitesimal fraction of what there is to know)

Earl said...

A nice, concise, airtight, elegant proof.