Double Edged

Thank you for reminding me why I hate math.

Um....if there is a discrete starting point for the set of natural numbers, is not this set finite in some sense? It's bounded on one side, but not on the other. Surely this makes a difference in comparing the two sets?

My head hurts, because I guess now I am saying that the set of natural numbers is both finite and infinite...

And why are we using spatial metaphors to describe nonspatial entities anyway?

funke,

First, all sets have a starting point. In the case of natural numbers and integers the starting point is zero. The set of natural numbers extends infinitely in one direction and the set of integers extends infinitely in both directions. This is why mathematicians say that the infinite set of integers is "larger" than the infinite set of natural numbers even thought it, also, is infinite. Basically, it is a mathematical ruse to decieve those who are potentially math minded into believing that one infinite can be greater than another infinite when in reality they are the same. If this ever "got out" then mathematics would be in trouble.

As to your second question (spatial metaphors describing nonspatial entities), I am much more capable of answering. Here the answer is simple: without spatial metaphors it would be impossible to know, understand or experience nonspatial entities.

Jared,

The set of natural numbers and the set of integers are equal in cardinality. In other words, the number of elements of one set is not larger than the number of elements in the other set. This can be demonstrated in constructing a mapping from one set to the other. This is proven by constructing a method of corresponding the element from one set to the other set.

However, not all infinite sets are equivelent in size. Consider the following two sets:

(a) The set of natural numbers.
(b) The set of real numbers between 0 and 1.

Each real number between 0 and 1 (not including 0 and 1) can be represented as a unique infinte decimal expansion. In the case of numbers with two equivelant decimal expansions, such as 0.5000... = 0.49999..., we choose the number ending in nines.

The proof that these sets are equivelant in size is done by a proof by contradiction. In this case, we assume that the two sets are equal in size and this will lead to a contradiction, showing that the assumption is false.

(1) Assume the set of counting numbers is equal in size to the set of real numbers in the interval between 0 to 1.

(2) Because of (1), we can arrange the numbers in a list:

1 -> r1 = 0 . a11 a12 a13 a14 ...
2 -> r2 = 0 . a21 a22 a23 a24 ...
3 -> r3 = 0 . a31 a32 a33 a34 ...
.
.
.
n -> rn = 0 . an1 an2 an3 an4 ...
.
.
.

The a's are digits (0 - 9). a23 is the digit for the r2 number (and the mapping from counting number 2) in the 3rd decimal position. This sequence of numbers of r1, r2, ... does not have to follow any particular order. For instance:

1 -> r1 = 0 . 2 4 2 9 8 ...
2 -> r2 = 0 . 1 3 1 4 5 ...
3 -> r3 = 0 . 4 1 5 7 3 ...
.
.
.

(3) We are now to construct a number x:

x = 0. b1 b2 b3 b4 b5 ...

where b1 = 1 if a11 is not equal to 1, otherwise it is equal to 2, b2 = 1 if a22 is not equal to 1, otherwise b2 = 2, etc.

Consider the example above,

x = 0. 1 1 1 ...

(3) Is x in the enumerated set of (2)? The answer is no. This is because for each digit b1, b2, b3, ... in x, that digit is different from the corresponding digit a11, a22, a33, ... in each of the numbers enumerated in the list of (2).

This contradicts that there can be a correspondence between the counting numbers and the set of real numbers between 0 and 1.

Therefore, the size of the set of the counting numbers are not equal in size to the set of real numbers.

So, in summary, the two examples of set you gave, then how can we say that {Sa}

Cardnality, which roughly corresponds with the notion of the size of the set, is strictly defined as the ability to map the elements completely from one set to the other and vise versa. In mathematical perlance, the sets are one-to-one and onto.

My first line in my summary was eaten. The two sets you listed are that same cardnality.

Chuck,

Yes, I heard and learned about the supposed difference between the two sets that you use as your example. A fellow student of mine who happens to be a math nut essentially showed me the "easy" way of demonstrating what you are saying. He said that given the set of real numbers between 0 and 1 it is possible to construct a number from that set that does not exist within that set. Basically this means that there is an infinite set which contains all real numbers and an infinite set which contains only the real numbers between 0 and 1. Since the set that contains all real numbers also contains the set of real numbers between 0 and 1, one is forced to conclude that the first set is "larger" than the second. It follows from this that if both sets are infinite (and they are) and one set is larger than the other, then there must be some infinite sets that are larger than other infinite sets. We still come up with the following:

{Rn} = all real numbers
{Sn} = all real numbers between 0 and 1

and

|Rn| = ∞
|Sn| = ∞

thus we can conclude {Rn} = {Sn} because they have the same cardinality. But we also know this isn't true intuitively because they cannot be put in a one-to-one ration with each other (as per your proof) so they can't have the same cardinality, yet they do.

The real problem here is with the mathematical assumption that the (to use your example) infinite decimal .5000... is equal to the infinite decimal .4999... when, quite obviously, it is not (one starts with a 4 and the other starts with a 5). Mathematicians make the mistake of believing that since there is virtually no difference between the two infinite decimals they, therefore, can be equated and/or substituted for one another without adverse affect. In my opinion, however, this should be a no-no along the sames lines as dividing by zero.

A set is a collection of objects or elements in no particular order. A proper subset is a set within a set that excludes at least one element of the set. Thus {A,B,D} is a proper subset of {A,B,C,D} but {A,B,C,D} cannot be a proper subset of itself. The cardinality of a set refers to the number of elements within a set. So the cardinality of {bear, car, apple} is 3. All this for the definition of an infinite set. A set is infinite iff the cardinality of at least one of its proper subsets equals the cardinality of the set as a whole. Consequently, the cardinality of integers is the same as that of natural numbers. There are just as many positive integers as there are positive and negative integers put together. The set of counting numbers, that is, whole positive integers, is just as large as the proper subset of those numbers that are divisible by 5. The same is true of a proper subset of numbers divisible by any counting number. The set of counting numbers, therefore, has an infinite number of proper subsets with a cardinality equal to the whole. Furthermore, the set of rational numbers, which consists of any number that can be expressed as a fraction, has the same cardinality as the set of counting numbers. The notion, then, that some infinite sets may have a larger cardinality than others has nothing to do with whether one is a proper subset of the other.

All of the number sets considered so far are proper subsets of the set of real numbers. But they have not included all the real numbers. For this, we need to include the irrationals. These cannot be expressed as fractions but are never ending non-repeating decimals. The set of all real numbers is infinite. The set of all real numbers between any two adjacent counting numbers is also infinite. Each of these sets has the same cardinality as the set of all real numbers. However, none of these infinite sets has the same cardinality as an infinite set containing only rational numbers (or rational numbers plus a finite subset of irrational numbers). An infinite set of real numbers containing an infinite proper subset of irrational numbers has a greater cardinality than an infinite set of real numbers that does not contain an infinite proper subset of irrational numbers. This is true because, after you think you've put them all in a one to one correspondence with the counting numbers, it's possible to think of more.

Try it. List all of the counting numbers in order. Now, match each of these numbers up with all of the numbers in the set of all real numbers using each of these only once. Both of these lists are infinite. If they have the same cardinality, then it will be impossible to think of a number that belongs in the second list that isn't already there. But it is possible. Make a decimal point and start at the tenths place. Insert any digit, 0-9, except for whatever digit is in the tenths place in the number corresponding to 1, or the first slot. Next, put any digit in the hundredths place except for whatever digit is in the hundredths place in the number corresponding to the number two slot. Keep going. Move down exactly one slot on the list and one decimal place to the right and write in any digit except for whatever digit is in the corresponding decimal place of the slot your working with. Do this until you've gone all the way down the list of counting numbers. You have constructed a number that belongs in the set of all real numbers. Nevertheless, it has not duplicated any of the numbers that are already on the list. It doesn't have its own slot. It can't be the same as the first number because the tenths place is different; the hundredths place is different from that in the second number; the ten-thousandths place is different than that in the fourth number. In fact, for whatever slot you may try to put this number, it will differ from the number already there in at least one of its digits. After populating an unrestricted set of real numbers whose cardinality is the same as the set of all counting numbers, there is still at least one number left. Even though both sets are infinite, it must be the case that the cardinality of the set of all real numbers is greater than that of the set of counting numbers.

Jared,

The mathematical definition of equivalent cardinality between two sets is there there exists a one-to-one and onto function that map maps all the elements from one set to the other and vise versa. Therefore your statement, "we can conclude {Rn} = {Sn} because they have the same cardinality." means that such a function exists. However, if you say, "But we also know this isn't true intuitively because they cannot be put in a one-to-one relation with each other" means you have intuitively denied what you just said. It turns out there is a function that exists that maps the Real Numbers in [0, 1] to the entire set of Real Numbers, and vise versa. That is why the Reals in the unit interval has the same cardinality as the entire set of Real Numbers.

By the way, I just did a google search for any interesting articles on infinite sets. Here is one that is oriented towards philosophers without a mathematics background: http://www.earlham.edu/~peters/writing/infapp.htm

Jared,

Regarding the assumption used in the proof that 0.5000... equals 0.499999... , that was placed in Cantor's proof to make sure that there function was truely a one-to-one and onto function. If that restriction was dropped, we actually don't have an one-to-one function, which means were the function is doing more than it needs to do. But we can drop that, and the theorem still holds. It's just that mathematical purists like to catch all the conditions and make a clean proof.

By the way, there is a proof for 0.5000... equals 0.4999... .

Kevin,

Your last paragraph doesn't make any sense to me. There can't be any numbers "left over" if there are an infinite number of numbers. It's like saying that there are an infinite number of numbers between 0 and 1 until you get to 1. If the set can get to 1 then it isn't infinite. If the set can't get to 1 then it has the same number of elements as any other infinite set, i.e. an infinite amount. You can state it the same way I've stated the other infinite sets:

{In} is the set of irrational numbers
{Rn} is the set of rational numbers

|In| = ∞ because there are an infinite number of elements
|Rn| = ∞ because there are an infinite number of elements

So if they both are equal to ∞ then why don't they equal each other? Because one set can contain an extracted number that doesn't exist thus giving it one more number than the other set? Both sets are infinite so it is possible to think of one more number in both sets; how does this mean one of the sets is larger? At any given point either set could be larger than the other, but as they are infinite they are equal in the number of their respective elements.

Chuck,

Of course there's a proof that .5000... equals .4999... That doesn't mean they really equal one another. It almost seems immoral to me to maintain that they do equal one another because the difference between the two is what mathematicians call negligible. Unfortunately for mathematicians a difference that is negligible is still a difference and, strictly speaking, if there is a difference then there can be no equality. Let's put it into a situational context:

Say the earth needs to be exactly .4999... miles away from the earth because any closer or any farther results in death for all living things. What mathematicians want to say is that we might as well round it up to .5000... because it's more convienient to work with that term and the difference between the two terms is such that there is "virtually" no difference. If in reality, however, the earth actually was .5000... miles instead of .4999... miles then we wouldn't even be having this discussion. So there is, in fact, a difference between the two terms and, depending on the circumstances, I would hardly call it negligible.

Jared,

It is not the case that mathematicians call the difference between these two numbers negligible. If they wish to be accurate, then they will call the difference non-existent. It seems to me that you're confusing a number in which the 9s go on for a really, really long time and then stop with this number in which they go on infinitely. In the first case, no matter how far they go, there is a real difference between this number and .5. In the second case, there is not. .4999.... and .5 are merely designations for the same value, much like 1/3 and .333..... are the same thing. If either of these are multiplied by three, the result can be represented as 3/3, 1/1, 1, or .999999..... Strictly speaking, there is no difference. I have posted my response concerning leftovers in infinite sets over here.

I must admit this discussion seems rather intense to me. As a result I'm not sure if 2+2=4 any longer and based on what I've just read I suspect it may not! :^)

I shall not even attempt to join the complexity of these posts other than to comment that I think Funke 'hit the nail on the head' with the statement, "The set of natural numbers extends infinitely in one direction and the set of integers extends infinitely in both directions." Therefore, at least in my mind, making one infinite set larger than the other.

Directional Infinity

Twist someone's arm enough and they'll eventually see your point. That, of course, doesn't make your point right even if it is mathematically demonstrable...

Hello,

A friend of mine pointed me to this blog post, and I'm encouraged to see this discussion taking place at Covenant.

I have accepted a position as a math professor at Covenant, and I will begin in the fall of 2007, Lord willing. If any of you will still be at Covenant at that time, I would look forward to meeting you.

If any of the participants and onlookers have not been satisfied with the explanations of Kevin and Chuck, feel free to email me, and I would be happy to attempt to clarify the situation.

Dr. Wilson,

I guess the only real or substantial problem I have with set theory is that it rubs my conception of "common sense" the wrong way and I prefer that common sense when given the choice between it and intellectualism/academia. Discovering a new number within a set after it's been mapped to another set seems dubious to me, kinda like pulling out a fifth ace from a "normal" deck then trying to claim that it was always there and we just needed a formula for uncovering it.

Jared,

Mathematics always does one of two things:

1) Confirm our common sense, thus strengthening our intuition and providing a method of applying it more effectively.

2) Confounding our common sense, thus . . .

Well, something must have failed, right? Not necessarily. Mathematics, like any legitimate study, reveals the nature of God. When #1 occurs, we see evidence of God's faithfulness in creating and orderly universe and giving us minds that can comprehend it. So what do we gain when #2 happens? What does that tell us? Assume that we check, and have verified that there is no error. Then we have a problem in mathematics and deduction, or in our common sense.

Now, where would we hope for the problem to be? Which problem would bring most glory to God?

Suppose the problem is with mathematics and deduction. Then our mathematical conclusions could never be more trustworthy than our intuition. Thus mathematics would never really advance our knowlegde, but only provide precision for what we already knew. In this case, mathematics is reduced to glorified accounting.

Suppose, on the other hand, the problem is with our common sense. What then? Well . . . our common sense was never sufficient for the physical universe, was it? The triumph of science is that we learn things that we never expected to be true. Moreover, we needed outside revelation to understand the nature of God, other than the most basic truths that are revealed in nature. Our common sense is not sufficient in theology. Moreover the more that we study God, the more we are aware of our inability to understand how his love and wrath fit together, etc.

When our common sense is confounded, we should study carefully, and make sure we aren't being fooled. We should strive to understand every side of the mystery, and make sure we feel the full weight of the tension. And then praise the surpassing wisdom of the God who has made an unknowably majestic universe and has chosen to reveal in mathematics some of the trancendence of his character. We should marvel at the smallness of our best intellectual efforts, and be grateful for the tremendous usefulness of our common sense in most cases.

Does this help?

Dr. Wilson,

It helps a little. I am, however, more inclined to err on the side of common sense than I am to err on the side of mathematics, even after proof checking. There's a lot of things that mathematics can do, especially depending on what type of mathematics you are working with. Therein lies the problem; a multiplicity of mathematical systems and what is mathematically true under one set of rules is not true under another set. So what truth does mathematics actually bring to the table if we understand truth as a reflection or finite representation of God's knowledge? The scientific industry as a whole is built upon assumptions that are relative to each respective field, so why should they be the primary influence upon what we consider truth? Quantum mechanics is a case in point from the math arena. Quantum mechanic formulas have been mathematically verified as much as general relativity formulas, yet they don't give the same mathematical "picture" of reality and, on a certain level, actually conflict/contradict each other. Bertrand Russell was right to ask "which picture is the 'real' table?" and this question exposes the main weakness in science, mathematics and logic.

When groups of fallen humans get together and start forming "laws" about a universe that is largely beyond our scope of comprehension, I tend to get a bit leary. Why? Because instead of sharpening our intuitions, an ardent reliance upon these institutions seems to spoil them. We start taking laws and theories for granted and end up digging holes that are difficult to climb out of. Now, I'm spoiled enough as it is being a sinner so the last thing I want to do is find myself jumping down one of those holes.

I was always under the impression that the "outside revelation" for understanding the nature of God were the most basic truths revealed in His creation. Mathematics and Science are simply tools (imperfect tools) that make the process a bit smoother when they are used properly. I can agree that the triumph of science is that we learn things we never expected to learn but we don't learn them at the expense of our common sense and/or intuition.

Jared,

Good questions. After all, chess is very complex, but we don't think that this is a reflection of God's transcendence, right? Because the rules of chess are arbitrary, and contrived for our entertainment.

But consider the case of the real numbers. What do we want with the real numbers anyway? You can look at it in two ways. You could consider the desire to count things, and then notice that negatives, fractions, irrational, and (sortof) transcendental numbers needed to be added in order for our operations of addition and multiplication to be nice. This perspective is useful, but I'll consider the more intuitive approach.

We want to be able to model distances. And the naive thought would be that the rational numbers would do that. But such a simple number as the hypotenuse of the unit square is not contained in the rationals. So we throw in the algebraic numbers, that is, the numbers that are roots of polynomials with rational coefficients. Still, we are not able to precisely express certain basic distances, such as the circumference of a circle.

What now? There is no easy fix. We need to fill in the gaps left by the algebraic numbers, and preserve our notions of addition and multiplication. I won't describe the formal construction of the real numbers here. But it turns out there is only one satisfactory way to do it. And if we don't do it, we have to give up on our goal of modeling distances.

And when we are done, it turns out that what we have thrown it was way more than what we had to begin with. (The algebraic numbers are countable, the non-algebraic numbers are uncountable.) In other words in completing the process of constructing a system to model distances, we have run into a case of infinite sets of different sizes!

So the surprising glory of this case is not that set theory can construct a system that has different sizes of infinite sets, but that this is found in the mathematical set that is the most basic and useful to humanity. The exciting reality is not just that crazy unintuitive things exist, but that these things cannot be avoided.

What are the most fundamental physical realities? Light, matter, and gravity would be good choices, right? And what do we understand about them? They are totally mysterious! God has seen to it that we have minds that are able to understand much about the universe (and mathematics) while being unable to fully understand any one thing. You are right, science is never dealing with reality. No scientific model is the truth, light is neither a wave or a particle. But yet, familiarity with the wave and particle theories of light provide much intuition into the behavior of light!

So what truth is brought when truth is dependent on the system that we start with? For one, we have the truth that no system is sufficient to address every question that could be asked. We also have the truth that systems that tried to answer simple questions are extremely complicated, though not inconsistent. We also see the principle (I haven't demonstrated this, but trust me) that when one pursues mathematical beauty, there is a feedback of applications to the physical world. (For example, imaginary numbers are perfect for modeling circuits.)

Don't get hung up about laws, the scientists aren't. They are in the business of throwing out the old laws when better ones are discovered. But they don't disparage the old laws, because they gave them the intuition to find the new laws. Good science improves intuition. I don't know if your common sense would tell you that a marble falls at the same speed as a bowling ball. But now that you know this, you have better intuition about other questions of mass and gravity.

So don't jump down the holes. Use the laws, but be ready to drop them.

I'm not sure exactly in what way you feel that Cantor's diagonal argument violates your common sense. Does it bother you that the proof involves several steps and you can't see the end from the beginning, or is it the result itself that is troubling? If the result that there are infinite sets with different cardinalities is uncomfortable, then I'm curious whether the existence of infinite sets bothers you, as it isn't exactly intuitive.

OK, again I hope that some of this was relevant, amidst ramblings.

By the way, will you be at Covenant in 07-08? I imagine that you are a philosopy student, is that correct?