In the event that some of you have never heard of the work of Georg Cantor and the idea that some infinities are larger than others (while some you would think are larger are actually the same size), consider checking it out below. It’s a nice introduction to the idea, teasing you to explore it further. Might be a good thing to blog about in more detail another time.

[By the way — on an oddly unrelated note: There may be an ad at the beginning for Mormonism (when I watched it, there was). I am not a Mormon, and I don’t endorse the ad, though the fellow in the commercial I saw seemed like a decent fellow. I firmly believe that if a dead fellow appears to you and tells you where you can find some cool golden plates you can use to found a new religion, you should just tell him that the God of the Bible commands you not to talk to the living impaired and move on. :) The video is below…]

They also have a video about 666, which I leave as an exercise for the student… and now I know (or think I do) why Cantor spent time in and out of mental institutions: not because of his math but because people didn’t accept it.

Hmmm. Aren’t our subdivisions of numbers though, in a sense arbitrary….whole numbers, integers, fractions, decimals? Wouldn’t infinity contain them all in the long run just as eternity contains all the befores, presents, and afters?

Teresa: Depends on what you mean by “infinity”. Not only are there infinities of different sizes, but some infinities contain other infinities within them. The set of whole numbers is infinite, yet it is smaller than the set of all numbers which is likewise infinite. The difference (so the video above points out) is that you can list the first but not the second. :D

Numbers aren’t arbitrary, they do have logical existence within the framework of mathematics. Each set of numbers has particular properties.

Cantor came up with the idea of the Absolute Infinite, which he took as the equivalent of God:

Howdy, Teresa! I wouldn’t consider the divisions arbitrary, but you are right in the sense that all of the real numbers can be found on the smooth, infinite number line. The question is whether or not all infinitely sized sets are the same size.

That question turns on how one measures the size of an infinite set and how one could compare those sizes. The most fundamental means is to match items up, one at a time, and if one set always has more than the other, then it is the larger set. Imagine, for instance, a civilization that can only count to three and no higher. If Albert has 4 goats and Bill has 5 goats, they won’t be able to count the goats to see who has more. But, clearly, they can see that for each goat Albert has, Bill also has a goat, yet when they are matched up, one-to-one, Bill has a goat that is unmatched while all of Albert’s are matched up. Since Bill always has a left over goat, he must have the bigger set. Even if we can’t count past three, we could still agree on that.

With infinite sets (e.g., All even numbers, All odd numbers, All integers, All fractions, All numbers bigger than 7, All irrational numbers, etc.), we find that we can line many up with each other in a one-to-one fashion such that every element in each set is perfectly matched with an element in the other set. Thus we see that the two sets are the same size. Obvious example: The odd numbers and the even numbers. Less obvious example: All the positive integers (the counting numbers) and all the positive integers bigger than 3. Even less obvious example: All the positive integers and all the integers (positive, negative, and zero). Crazy, not-at-all obvious example: All the positive integers and all the rational numbers (fractions). In each of those cases (the last one is discussed a bit in the video), each of those infinitely large sets of numbers is the same size as the other — and, in fact, all of those sets are the exact same size.

When things change is when we get to the irrational numbers: those numbers that cannot be expressed as a fraction (that is, a plain-old ratio of integers, like 2/3 or -7/5 or even 0/18). Irrational numbers–like pi (π) or the square root of 2–cannot be expressed as fractions or decimals that end in repeated strings (such as 6/11 = 0.5454545454…). And, it just so happens that when you try to match these up with the other sets we’ve mentioned so far, you always have left over irrational numbers that aren’t matched up, even though all of your other set is completely used up. Consequently, the size of the infinite set of irrational numbers must be bigger than the size of the infinite set of, say, the counting numbers or the rational numbers, etc. (This last comparison is one the video was trying to help suggest.) In this way, the rational numbers are like an infinite number of bright stars, and the irrational numbers are the blackness that fills the space between in a somehow-more-infinite sort of way.

Thus, we come to the idea of infinities of differing sizes. I hope this makes sense! I might try and make a full post on it at a later time–it really is one of my favorite ideas in mathematics.

Very nice explanation, Mr. Smith. I have to take your word on an element here and there, but given that everything else makes sense to me. When you do your longer explanation, if not sooner, would you mind comparing all the whole numbers on an infinite Cartesian plane with all the numbers of any kind on an infinite Cartesian plane?

This was a fun video. I liked the way he explained it in simple terms.

The subject of sets and infinities is closely related to symbolic logic; which your are no doubt familiar with. (I’m logic and philosophy buff, but the high end math stuff beyond me).

I have a question for you. While math shows us the possibility of infinity, it nonetheless deals in the material world. Infinity should include the should include both the material and spiritual universe, but math only expresses infinite possibilities within the limited (material) universe.

In short, does math eventually break down when we talk about infinity?

Yeah, I know my question sounds a little odd. I’m reacting to an article that I read. It said that math does provide an explanation for the creation of the physical universe without the existence of God. Something about fluctuations between space and time, which mathematically do exist. Yet I wonder how did those mathematical laws come into existence, and when did that happen. If the physical universe did not exist, then how could the math expressing the physical universe exist? Do you know what I mean? Something is fishy in Denmark.

On an unrelated note: “Diagnosis Christianity” = BRILLIANT Presentation! Mr. Smith, this is one of the greatest programs I have ever seen. You hit the proverbial “nail on the head”. Well done!

John Wheeler: I will try to explain such things more clearly in the longer post. As for whole numbers on the Cartesian plane versus numbers of any kind (I’ll have to assume you mean real numbers at this point or even complex numbers; there are lots of kinds!), I’m not sure what you mean. The Cartesian plane is two dimensional and its points identified by pairs of numbers, so do you mean lattice points (points where the x– & y-coordinates are integers versus points where the coordinates are any real number? I guess I’m asking: Are you wanting me to compare the collection of points with integral coordinate pairs with the entire collection of points? If so, the comparison is identical. The lattice points are a smaller infinity (in this case, a countable or listable one).

steve: No, math doesn’t break down when we talk about infinity. Actually, it’s quite the opposite in that mathematics helps us to establish a language to discuss infinity sensibly (or at least something approximating sensibly). However, based on your second comment, that is irrelevant to your real concern, which is a concern I share: Math and physical laws are often wrongly used by atheists as a “God substitute.” However, 1 + 1 = 2 isn’t an act of creation, it is a statement of a fact. Ditto for F = ma. Numbers are inherently impotent in the world in all the important senses under discussion, here, and have no causative powers whatsoever. They represent abstract objects, not causative agents–and that is a vital distinction.

I’ve seen physicists admit that for many of their “God-less” creation stories to be true, laws must have been in existence before those things the laws are meant to govern. When they say that, you realize you’ve won the discussion, as there is nothing outside of metaphysics that will allow you to discuss a universe that is nothing, absolutely nothing, but laws.

If you would like to see such contortions (a real pleasure, I admit it; I like winning) from one of the big names in physics today, read Alex Vilenkin’s discussion of the possibility in his Many Worlds In One: The Search for Other Universes. He suggests quantum tunneling from “nothing” as a means of “creating” the universe, but he recognizes that the nothing could not have been truly nothing but that the laws of the universe must have pre-existed the universe itself, which leads him to a very special place and quite an admission:

“The picture of quantum tunneling from nothing raises another intriguing question. The tunneling process is governed by the same fundamental laws that describe the subsequent evolution of the universe. It follows that the laws should be ‘there’ even prior to the universe itself. Does this mean that the laws are not mere descriptions of reality and can have an independent existence of their own? In the absence of space, time, and matter, what tablets could they be written upon? The laws are expressed in the form of mathematical equations. If the medium of mathematics is the mind, does this mean that mind should predate the universe?

“This takes us far into the unknown, all the way to the abyss of great mystery. It is hard to imagine how we can ever get past this point. But as before, that may just reflect the limits of our imagination.”

It does, indeed, reflect some limits, but not just of imagination. His reasoning has taken him to a door he refuses to open, though the path leads straight to it and through it. Let’s not let them impose such limits on us. :)

Thanks, Mr Smith. Your statement “[numbers] represent abstract objects, not causative agents..” was a good point. And I took note of the italicized quote you mentioned. Thanks again!

John Wheeler: The real numbers are definitely not listable. They include the irrational numbers, which are not listable. And if you are only comparing sets of numbers, then the plane is an inappropriate geometric space. You’re talking about a comparison for which only one dimension is necessary, so the infinite number line would be your working space, not an infinite Cartesian xy plane.

That said, the real numbers form a non-countable (or non-listable) infinite set, while the integers are a countably infinite set. So the size of the real numbers as a collection represents a larger infinity than that associated with the integers.

Teresa’s comment, “Wouldn’t infinity contain them all in the long run just as eternity contains all the befores, presents, and afters?”, makes more sense than anything else I’ve read here. Sorry guys.

Again, TeapotTempest – it depends on how you define “infinity”. We’ve been taught to look at it in the broadest possible terms (which certainly works for everyday and biblical use). It’s on the mathematical level that “infinity” has more than one, if this is the right word, meaning. Some infinities, counterintuitive though it might seem, really are larger than others. That’s why it took such a long time for Cantor’s peers to accept his reasoning. It seemed at first to be, well, crazy. (And I can understand why the kind of opposition he received would drive him crazy.)

And that brings me, Mr. Smith, to my mortification (I am not exaggerating) at not seeing the obvious in my own question. You shouldn’t have needed to point out that to examine the problem, a two-dimensional plane isn’t necessary when a one-dimensional line will do. It just is easier for me to see the problem in my head in such terms. But I once was good at mathematics. Now it seems I can’t even remember the definition of real numbers. Shall I let you shoot me now (to paraphrase Bugs Bunny and Daffy Duck), or wait until I get home? :P With a Vulcan cannon, if they still make such things for military aircraft.

And just one other thought: Cantor looks and writes like a fellow ENFP, on first impression. If so, then no wonder he got overwhelmed by his opposition. He couldn’t help but take it personally, much more than most people, and quite despite himself. What he went through reminds me of my own reactions to life and people, all too vividly.

TeapotTempest: Well, you’re half way there then, which isn’t bad. :) It is simply that the fact that collections with fewer sorts of things–such as, say, the counting numbers, only–are also infinite presents us with a question: Are those infinities inherently the same in nature/size/etc. or is there a difference? That’s all that’s under discussion, here.

Brady Haran’s http://www.youtube.com/user/numberphile video where he tries (and succeeds!) in throwing a yahtzee was hilarious. Did you notice his counting mechanism? A real high technology piece of equipment! Tooo much! A fun piece to watch. Thanks for bringing that link to my attention.

Cool video!

They also have a video about 666, which I leave as an exercise for the student… and now I know (or think I do) why Cantor spent time in and out of mental institutions: not because of his math but because people didn’t accept it.

Hmmm. Aren’t our subdivisions of numbers though, in a sense arbitrary….whole numbers, integers, fractions, decimals? Wouldn’t infinity contain them all in the long run just as eternity contains all the befores, presents, and afters?

Teresa:Depends on what you mean by “infinity”. Not only are there infinities of different sizes, but some infinities contain other infinities within them. The set of whole numbers is infinite, yet it is smaller than the set of all numbers which is likewise infinite. The difference (so the video above points out) is that you can list the first but not the second. :DNumbers aren’t arbitrary, they do have logical existence within the framework of mathematics. Each set of numbers has particular properties.

Cantor came up with the idea of the Absolute Infinite, which he took as the equivalent of God:

http://en.wikipedia.org/wiki/Absolute_Infinite

Beyond saying that, I am

wayout of my league. :DHowdy, Teresa! I wouldn’t consider the divisions arbitrary, but you are right in the sense that all of the real numbers can be found on the smooth, infinite number line. The question is whether or not all infinitely sized sets are the same size.

That question turns on how one measures the size of an infinite set and how one could compare those sizes. The most fundamental means is to match items up, one at a time, and if one set always has more than the other, then it is the larger set. Imagine, for instance, a civilization that can only count to three and no higher. If Albert has 4 goats and Bill has 5 goats, they won’t be able to count the goats to see who has more. But, clearly, they can see that for each goat Albert has, Bill also has a goat, yet when they are matched up, one-to-one, Bill has a goat that is unmatched while all of Albert’s

arematched up. Since Bill always has a left over goat, he must have the bigger set. Even if we can’t count past three, we could still agree on that.With infinite sets (e.g., All even numbers, All odd numbers, All integers, All fractions, All numbers bigger than 7, All irrational numbers, etc.), we find that we can line many up with each other in a one-to-one fashion such that every element in each set is perfectly matched with an element in the other set. Thus we see that the two sets are the same size. Obvious example: The odd numbers and the even numbers. Less obvious example: All the positive integers (the counting numbers) and all the positive integers bigger than 3. Even less obvious example: All the positive integers and all the integers (positive, negative, and zero). Crazy, not-at-all obvious example: All the positive integers and all the rational numbers (fractions). In each of those cases (the last one is discussed a bit in the video), each of those infinitely large sets of numbers is the same size as the other — and, in fact, all of those sets are the exact same size.

When things change is when we get to the irrational numbers: those numbers that cannot be expressed as a fraction (that is, a plain-old ratio of integers, like 2/3 or -7/5 or even 0/18). Irrational numbers–like pi (π) or the square root of 2–cannot be expressed as fractions or decimals that end in repeated strings (such as 6/11 = 0.5454545454…). And, it just so happens that when you try to match these up with the other sets we’ve mentioned so far, you

alwayshave left over irrational numbers that aren’t matched up, even though all of your other setiscompletely used up. Consequently, the size of the infinite set of irrational numbers must be bigger than the size of the infinite set of, say, the counting numbers or the rational numbers, etc. (This last comparison is one the video was trying to help suggest.) In this way, the rational numbers are like an infinite number of bright stars, and the irrational numbers are the blackness that fills the space between in a somehow-more-infinite sort of way.Thus, we come to the idea of infinities of differing sizes. I hope this makes sense! I might try and make a full post on it at a later time–it really is one of my favorite ideas in mathematics.

Very nice explanation, Mr. Smith. I have to take your word on an element here and there, but given that everything else makes sense to me. When you do your longer explanation, if not sooner, would you mind comparing all the whole numbers on an infinite Cartesian plane with all the numbers of any kind on an infinite Cartesian plane?

This was a fun video. I liked the way he explained it in simple terms.

The subject of sets and infinities is closely related to symbolic logic; which your are no doubt familiar with. (I’m logic and philosophy buff, but the high end math stuff beyond me).

I have a question for you. While math shows us the possibility of infinity, it nonetheless deals in the material world. Infinity should include the should include both the material and spiritual universe, but math only expresses infinite possibilities within the limited (material) universe.

In short, does math eventually break down when we talk about infinity?

Yeah, I know my question sounds a little odd. I’m reacting to an article that I read. It said that math does provide an explanation for the creation of the physical universe without the existence of God. Something about fluctuations between space and time, which mathematically do exist. Yet I wonder how did those mathematical laws come into existence, and when did that happen. If the physical universe did not exist, then how could the math expressing the physical universe exist? Do you know what I mean? Something is fishy in Denmark.

On an unrelated note: “Diagnosis Christianity” = BRILLIANT Presentation! Mr. Smith, this is one of the greatest programs I have ever seen. You hit the proverbial “nail on the head”. Well done!

Thanks, Steven.

John Wheeler:I will try to explain such things more clearly in the longer post. As for whole numbers on the Cartesian plane versus numbers of any kind (I’ll have to assume you mean real numbers at this point or even complex numbers; there are lots of kinds!), I’m not sure what you mean. The Cartesian plane is two dimensional and its points identified by pairs of numbers, so do you mean lattice points (points where thex– &y-coordinates are integers versus points where the coordinates are any real number? I guess I’m asking: Are you wanting me to compare the collection of points with integral coordinate pairs with the entire collection of points? If so, the comparison is identical. The lattice points are a smaller infinity (in this case, a countable or listable one).steve:No, math doesn’t break down when we talk about infinity. Actually, it’s quite the opposite in that mathematics helps us to establish a language to discuss infinity sensibly (or at least something approximating sensibly). However, based on your second comment, that is irrelevant to your real concern, which is a concern I share: Math and physical laws are often wrongly used by atheists as a “God substitute.” However, 1 + 1 = 2 isn’t an act of creation, it is a statement of a fact. Ditto forF=ma. Numbers are inherently impotent in the world in all the important senses under discussion, here, and have no causative powers whatsoever. They represent abstract objects, not causative agents–and that is a vital distinction.I’ve seen physicists admit that for many of their “God-less” creation stories to be true, laws must have been in existence before those things the laws are meant to govern. When they say that, you realize you’ve won the discussion, as there is nothing outside of metaphysics that will allow you to discuss a universe that is nothing, absolutely nothing, but laws.

If you would like to see such contortions (a real pleasure, I admit it; I like winning) from one of the big names in physics today, read Alex Vilenkin’s discussion of the possibility in his

Many Worlds In One: The Search for Other Universes. He suggests quantum tunneling from “nothing” as a means of “creating” the universe, but he recognizes that the nothing could not have been truly nothing but that the laws of the universe must have pre-existed the universe itself, which leads him to a very special place and quite an admission:“The picture of quantum tunneling from nothing raises another intriguing question. The tunneling process is governed by the same fundamental laws that describe the subsequent evolution of the universe. It follows that the laws should be ‘there’ even prior to the universe itself. Does this mean that the laws are not mere descriptions of reality and can have an independent existence of their own? In the absence of space, time, and matter, what tablets could they be written upon? The laws are expressed in the form of mathematical equations. If the medium of mathematics is the mind, does this mean that mind should predate the universe?“This takes us far into the unknown, all the way to the abyss of great mystery. It is hard to imagine how we can ever get past this point. But as before, that may just reflect the limits of our imagination.”It does, indeed, reflect some limits, but not just of imagination. His reasoning has taken him to a door he refuses to open, though the path leads straight to it and through it. Let’s not let them impose such limits on us. :)

Mr. Smith:I meant, on an infinite x-y plane, all integers verses all real numbers. And if they’re both listable, then that answers my question.I love the rest of what you deal with. Good show, as they say (or perhaps used to say) across the pont. :D

Thanks, Mr Smith. Your statement “[numbers] represent abstract objects, not causative agents..” was a good point. And I took note of the italicized quote you mentioned. Thanks again!

John Wheeler:The real numbers are definitelynotlistable. They include the irrational numbers, which are not listable. And if you are only comparing sets of numbers, then the plane is an inappropriate geometric space. You’re talking about a comparison for which only one dimension is necessary, so the infinite number line would be your working space, not an infinite Cartesianxyplane.That said, the real numbers form a non-countable (or non-listable) infinite set, while the integers are a countably infinite set. So the size of the real numbers as a collection represents a larger infinity than that associated with the integers.

I hope this helps!

Mr. Smith:Yes, that helps on all fronts.Teresa’s comment, “Wouldn’t infinity contain them all in the long run just as eternity contains all the befores, presents, and afters?”, makes more sense than anything else I’ve read here. Sorry guys.

Again,

TeapotTempest– it depends on how you define “infinity”. We’ve been taught to look at it in the broadest possible terms (which certainly works for everyday and biblical use). It’s on the mathematical level that “infinity” has more than one, if this is the right word, meaning. Some infinities, counterintuitive though it might seem, really are larger than others. That’s why it took such a long time for Cantor’s peers to accept his reasoning. It seemed at first to be, well, crazy. (And I can understand why the kind of opposition he received would drivehimcrazy.)And that brings me,

Mr. Smith, to my mortification (I am not exaggerating) at not seeing the obvious in my own question. You shouldn’t have needed to point out that to examine the problem, a two-dimensional plane isn’t necessary when a one-dimensional line will do. It just is easier for me to see the problem in my head in such terms. But I once wasgoodat mathematics. Now it seems I can’t even remember the definition of real numbers. Shall I let you shoot me now (to paraphrase Bugs Bunny and Daffy Duck), or wait until I get home? :P With a Vulcan cannon, if they still make such things for military aircraft.And just one other thought: Cantor looks and writes like a fellow ENFP, on first impression. If so, then no wonder he got overwhelmed by his opposition. He couldn’t help but take it personally, much more than most people, and quite despite himself. What he went through reminds me of my own reactions to life and people, all too vividly.

TeapotTempest:Well, you’re half way there then, which isn’t bad. :) It is simply that the fact that collections with fewer sorts of things–such as, say, the counting numbers, only–are also infinite presents us with a question: Are those infinities inherently the same in nature/size/etc. or is there a difference? That’s all that’s under discussion, here.Brady Haran’s http://www.youtube.com/user/numberphile video where he tries (and succeeds!) in throwing a yahtzee was hilarious. Did you notice his counting mechanism? A real high technology piece of equipment! Tooo much! A fun piece to watch. Thanks for bringing that link to my attention.