Tag Archives: Cantor

Set Theory

Grouping objects, whether they are tangible objects such as cars, books or animals or intangible objects like colors or numbers is not hard to do. There are in fact many ways to do so, the most familiar being the way that most elementary students learn. Venn diagrams are usually one of the first things we learn about set theory. Basic Venn diagrams are normally drawn as circles that overlap. If the first circle (we’ll call it “A”) represents the group (set) of insects that sting and the second circle (circle “B”) represents the group (set) of insects that fly then all of the insects that both fly and sting would be represented by the overlapping part of the circles.

When done on math, grouping objects is known as Set Theory. Sets are represented in a different way but it is still the same concept. You can define a set to be a group of actual object or you can define a set to by a specific rule such as “set A contains all even numbers”. The objects in a set are called “elements” and the operations of sets are quite simple, the most common being the union, intersection and difference. The union is simply the set of elements that contain any elements of set A, B or both. The intersection is the set of elements that set A and B both have in common while the difference is the set of elements that are in A but not in B. Using Venn diagrams as an example, if we highlight the areas of a circle that is the union of A and B then both circles would be completely highlighted. For the intersection the area of the circles that overlap would be the area that is highlighted and for the difference the area of the circle that would be highlighted is the part of circle A that is not overlapping with B.

Venn diagram representation of a union.

Venn diagram representation of a union.

Venn diagram representation of an intersection.

Venn diagram representation of an intersection.

Venn diagram representation of a difference.

Venn diagram representation of a difference.

Although named for him, John Venn did not invent these diagrams; logicians have used them for centuries. It was common in the 19th century to use Euler diagrams (Eulerian circles). Euler diagrams consisted mainly of circles within circles and occasionally circles by themselves. As an example, if the outer circle represented insects that sting, then the circle inside of that would represent insects that both sting and fly. A completely separate circle would represent something that neither flew nor stung. John Venn felt that theses diagrams were inadequate and reverted back to a diagram that has been used throughout history. Since Venn formalized these diagrams and was the first to generalize them, they were later named after him.

It is interesting to note that the original purpose for Venn diagrams was not set theory but rather symbolic logic. Symbolic logic uses symbols rather than words in order to remove the ambiguity that some words tend to have. When using abstract symbols rather than familiar words, it is harder to see the truth of a statement. Venn diagrams helped greatly with this. In symbolic logic you have two premises and a conclusion.

Most mathematical topics normally develop through the collaboration of many mathematicians, but a single mathematician, Georg Cantor, founded set theory in the late nineteenth century. There are many different subfields of set theory including Combinatorial set theory, Descriptive set theory, Fuzzy set theory and Rough set theory, but the one that is most widely known among mathematicians is Zermelo-Fraenkel set theory (ZFC). ZFC was originally developed in an attempt to rid set theory of paradoxes such as Russell’s Paradox, discovered in 1901 by Bertrand Russell. Russell’s Paradox can be stated as such: Let set R be the set of all sets that are not members of themselves. If R is not a member of itself, then by definition it must contain itself. But this contradicts its own definition of being the set of all sets that are not members of themselves.

Because every mathematical object can be viewed as a set, any mathematical statement can be written in set theory notation and therefore any mathematical theorem can be derived using ZFC set theory. The reason ZFC set theory is so well known among mathematicians is, because of this, it is at the foundation of almost all modern mathematics.

http://www.britannica.com/EBchecked/topic/625445/John-Venn

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

http://plato.stanford.edu/entries/set-theory/#SetTheFouMat

http://www.encyclopediaofmath.org/index.php/ZFC

http://www-history.mcs.st-and.ac.uk/HistTopics/Beginnings_of_set_theory.html

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

My infinity is bigger than your infinity

When I was a child, I purposely found something to think about to help me fall asleep. Usually I picked cartoons or super powers, but sometimes things just came into my head, like it or not. What was the worst? Thinking about heaven. At first, heaven seems all right. There is a lot to do, gold everywhere (though no purpose for it), people are nice (it’s a prerequisite), you get to see most of your family, and there is plenty to eat (though no one is ever hungry). Anyway, I start thinking about FOREVER.

pic1cropAt first, it is just a sensation; a weird sensation like tingling and falling and nothingness. It is not a sensation that I can make sense of really because forever doesn’t really make sense, at least not to a 10 year old. I try to get away from forever but forever is a huge part of the definition of heaven. Then, the opening credits of the Twilight Zone, with the music, and starry sky, usually appear. Fade to myself standing, looking at heaven, in the dressing room mirrors of infinity. You know, when dressing rooms have those three mirrors that are angled just perfectly so the images are smaller and smaller replicas of one another, on and on, into infinity. This picture, and thoughts of the foreverness of heaven, kept me up at night as a child.

I am glad to say that forever no longer keeps me up at night. While I still find no comfort in the foreverness of heaven, the lack of a middle ground between forever and my time on earth is what usually keeps me up at night now. However; I still can’t stand it when mirrors are angled that way. It creeps me out, and I can’t help but wonder if there is an end, or if I can find a flaw from one image to the next. In my opinion, we are not meant to look into infinity like that, squarely.

When beginning to pursue mathematics, I thought math might clarify, or in some way define, forever (or as adults call it, infinity). On the contrary, Math has actually made it stranger. Theories in math have shown numerous types of infinity, and infinities within infinities, and sizes of infinities, and calculations of infinity. None of this brings me any comfort, except to say that we obviously don’t have this figured out yet because that is just not possible. Infinity is infinity, and it is very large, incalculable and non-denumerable, and there is only one kind; it is called forever. Heaven can only exist in one, all-encompassing infinity.

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When reading A History of Mathematics, I read about Zeno’s paradox. That led to an internet search, and then to Numberphile. I watched the video, accepted the idea, and left it alone. The solution seemed reasonable enough. Later in the semester, I was required to do a research project. By some unknown scheme, we picked Georg Cantor, whom I had never heard of. If you haven’t either, he is the creator of set theory but also perhaps the mathematical or scientific father of infinity. You just can’t shake things off in life. They follow you.

My research for that project led me to question the mathematical view of infinity. Let me start by saying, I know very little of Math’s view of infinity. It seems to be an infinite topic. This is where I am in my understanding – so please comment, post, reply, educate me, and critique my understanding. Calculus one is a prerequisite for the course, and being a rule follower, I have that. So, I had experience computing limits to infinity. That is relatively easy. BUT, those are just numbers. They aren’t real things. Numbers aren’t real. So, of course I could compute the infinity of something that isn’t really real. What numbers represent is real; like Zeno’s paradox. Zeno’s paradox applies numbers to something real – something actually happening in the world (theoretically). In other words, when I take the limit of a sequence that goes to infinity, it has no relation to time or space. It is just numbers. But, if I were taking the limit of Zeno’s paradox to see how far Zeno actually travels, or to find the time it takes to travel, or to see if he can ever catch the turtle, I would have to do so in relation to time and space. When I do that, the exact opposite answer occurs. Zeno will never catch the turtle. That mathematics isn’t computing real infinity or perhaps all of infinity is perhaps echoed by the Numberphile narrator when he asks, “What I want to ask a physicist is, can you divide space and time infinitely many times?” Similarly, Kelly MacCarthur wonders in the Calculus 2 video used for online math courses, “Can I take infinitely many steps?”

However, if all of space and time existed at one instant, forever, then Math has it right. It could calculate the infinite because it occurs all at once. There is no sequence, event after event – in essence, no time or space really because it is all at once, everywhere. Yes, there are scientific theories, philosophies, and religions which believe this is the case. Of course, this idea is contrary to most people’s understanding of infinity. Whenever math instructors talk about infinity, they always say, “Infinity is only a concept. It is not a number.” Yes, it is only a concept but is it also something real? If it is only a concept then why are we computing something real that is a concept? Why would we bother to compute a concept? It seems like Math is walking a funny line here.

Math has worked something out though. I’m just not sure what it is. Math is summing an infinite process (as if infinity happened to end). Obviously, Math’s understanding of infinity has proven useful in mathematical calculations and many practical applications. To paraphrase others before Cantor, “It works. So, no need to define it. It works.” So, Math has worked something out about infinity but what has Math worked out, and is it really infinity?

pic3cropMathematicians always like to joke about engineers rounding numbers to 3 or 4 places because it doesn’t really matter to engineering after that, but is mathematics rounding off infinity or at least only capturing some aspect of infinity? After all, how can there be different types of infinity? My preferred illustration for the existence of multiple infinities is from Galileo. Galileo used a thoughtful but intuitive approach to understand infinity. He drew a circle. Then, he drew an infinite number of rays from the center of the circle. These rays filled up the space inside the circle. But then, he drew a larger circle around the smaller one and extended those rays to the larger circle. Though he drew as many rays as possible (an infinite number perhaps), the infinite number of rays did not fill up the larger circle; there were spaces between the rays. This led him to believe that first infinity was not large enough for the second circle; not even close. He would need another size of infinity to fill up the larger circle. [BAM! PHH! Did your mind just explode?] It is important to note that intuitively, his illustration makes sense. However, with today’s current understanding of infinity and better ability to calculate infinity, we now know that the infinity in the smaller circle leaves no space between the rays when extending to a larger circle. But, I liked his intuitive approach. Though intuition seems to be severely lacking when it comes to infinity.

References:

Dangerous Knowledge: http://topdocumentaryfilms.com/dangerous-knowledge/
Georg Cantor His Mathematics and Philosophy of the Infinite by Joseph Warren Dauben
TML: The Infinities In Between (1 of 2): http://www.youtube.com/watch?v=WihXin5Oxq8
TML: The Infinities In Between (2 of 2): http://www.youtube.com/watch?v=KhgNiqI-bt0
Infinite Series: http://stream.utah.edu/m/dp/frame.php?f=f55f900bec01a3106121
Zeno’s Paradox – Numberphile: http://www.youtube.com/watch?v=u7Z9UnWOJNY

My new bumper sticker.

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