What happens to the volume of a sphere in higher dimensions? To answer this question, we will focus our attention on the unit -sphere in Euclidean space. That is the sphere in (-space) with radius centered at the origin.

For example, in , the unit sphere is the collection , which satisfies the equation . This may seem a little strange, but this matches exactly the pattern in higher dimensions.

In we have the unit circle satisfied by the equation .

In we get an “hollow globe” satisfied by .

This pattern continues to where the unit sphere here is given by the equation .

As a side note: it is common to think of the “filled” versions of these spheres. Algebraically, this means replacing the “” symbol with the “” symbol. If we do that, for example, in -space we get the interval and in -space we get the unit disk. This makes visualizing volume a little easier.

But what exactly is **volume**? Put simply, the **volume **of an -dimensional** **object is the amount of -dimensional *stuff *we can put *inside* the object. For us, for example, it is the amount of stuff we can fit inside a box, a glass, a backpack, etc.

We saw in a previous post, the volume of a unit -sphere with radius is given by

.

Since we are concerning ourselves with the unit sphere, we will use to represent . Using this formula, we can quickly generate a few of these values:

Wait.. what? When we moved from 5 dimensions to 6 dimensions we decreased in volume. Does this pattern continue? Let’s plot.

Curious, the plot clearly shows that by the time we reach dimension 20, have almost 0 volume. In fact, the volume of the unit sphere tends to 0 as dimension increases, i.e. we have

.

The proof is an excellent exercise in limits that we omit here. The fact alone is baffling enough. If we were -dimensional beings, we certainly wouldn’t be storing our things in spheres. Is there something we could store our -dimensional things in?

Let’s consider the unit circle one more time. We can circumscribe it with a square with side length . In the plane, this square has corners . The square has area square units.

Similarly, in -dimensions, we can circumscribe the unit sphere with a cube of side length . This cube has corners and . This cube has volume cubic units.

We can continue this pattern and circumscribe an -dimensional unit sphere with an -dimensional cube of side length 2. This cube will have -dimensional volume of . Then we clearly see that as the dimension increases, the volume of the cube increases. That is to say

.

This phenomenon just got weirder. Intuitively, from what we know of and dimensions, the volume of a sphere and it’s circumscribed cube should be on comparable levels. This is clearly not the case. One can gather from this information that volume in -dimensional spaces is concentrated in corners since the sphere and cube are somehow touching on all sides, yet the sphere has exponentially less volume in higher dimensions.

What doesn’t quite come through here is the fact that it isn’t really feasible to compare -dimensional area and -dimensional volume since square units are not the same as cubic units. Moreover, by adding a dimension, one increases the ability to store more lower dimensional “stuff”. For example, there are infinitely many -dimensional unit circles contained in the -dimensional sphere. This works similarly for other dimensions.

For any budding topologists/analysts/geometers one natural direction for this is the fact that the “shape” of the unit sphere changes depending on the space we are in. For example, the typical object we think of when we think of a sphere is the unit sphere for $L^2$. The corresponding circumscribed cube is actually the “unit sphere” for a space called . In a future post, we will give an accessible (hopefully) introduction to these spaces and the behavior of the volume of their unit spheres.

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Let’s begin with an important question: What is the value of the following integral:

.

This is known as a Gaussian integral, and is related to one of the most important concepts seen in basic statistics, the Normal (Gaussian) Distribution

To answer this question, we will square it and compute the resulting integral:

.

By renaming the variables we get

Now we rewrite as a double integral

Which quickly gives

We are now in a position to make the change to polar coordinates, which we have briefly discussed before:

So that

By making the simple substitution, , we get

.

This reduces to

.

So finally, we’ve answered the question and see

.

But what does this have to do with the volume of a sphere? Maybe it was slightly overlooked, so let’s do it in general. Notice in the switch to polar coordinates above, the angular piece played no large part of the integral. That will remain true in general. Also notice that by squaring the Gaussian, we were pushed into two dimensional polar coordinates. The same is true for powers of . Let’s see.

.

This gives us the -fold integral (don’t be intimidated by what follows, we’ll clean it up immediately with one move)

.

Now we make the switch to (hyper) spherical coordinates. If you are not familiar with this subject, just know there is a radial piece and angular pieces that we wrap all together. Which allows us to write

We can condense this even more by letting . It should be noted that is actually the surface area of the -sphere, see above for a visual in 3-space. We don’t need to know the actual value of right now, it will actually fall right out in the end. Making the proper substitution,

.

Although the integral remaining looks slightly terrifying, it almost has the form of a Gamma integral. To reach this form we make the substitution, so that and

.

If we recall our original question, this gives us

.

But we already know the left hand side above as . So we have

.

Recall that is the surface area of our -sphere. To get the volume we simply integrate against our radius, say .

We can reduce this using the fact .

.

This is the volume of a sphere in -dimensions with radius . In the next post, we will explore an interesting consequence of this formula that is, for most, unintuitive.

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Archimedes had been contracted by the tyrant King Heiro II to find a way to check if his crown was pure gold. The King had suspected that a bit of silver had been used in making the crown. Archimedes’ first solution, if the King was so sure, was to crush the crown into a small cube where the volume could easily be measured and compared with the mass. Density (the ratio of mass to volume) was a well known concept, as was the fact that gold is more dense than silver. The King would not allow this. Archimedes, in a huff, took to the bath where the famed moment occurred. Eureka! He had discovered that one can measure the volume of the object simply by measuring the volume of water that had been displaced.

This meant that he could take the crown and submerge it, then find the volume of the crown my measuring the displacement of water.

While the crown does not appear in the work of Archimedes, it is the most well known legend of the man.

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What spurs the advancement of civilization? This is easily a very complex question, however the simple answer we will use is *technology*. Civilizations advance as their technologies advance. Where once there were hunters and gatherers, tools and other knowledge allowed for the possibility of crop cultivation. This sort of technological innovation can be seen during every major change of civilization. At the heart of this innovation sits mathematics. But we couldn’t have gotten to civilization as we know it today just with basic geometry and counting.

So, what happened? Abstraction is an important key in the advancement of mathematics. Back to our students, as they progress through learning how numbers interact, they eventually reach the true first level of abstraction. Instead of learning how numbers interact with each other through examples, students learn how number interact with each other through ideas. It is at this point, our students have reached **algebra**. In algebra we focus on the ideas of how numbers interact with each other. We learn things like the distributive property, polynomials, graphing, solving equations, etc. These theorems that are developed and taught in algebra classes, show the students that numbers are predictable. Now instead treating each multiplication problem independently, they get to see that it is truly the same across the board. It is at this point our story splits and follows those students that delve further into the depths of mathematics.

Our math loving students will next make it in to **calculus**, where they have their first true dance with the real numbers. But the ideas of calculus are not restricted to just the real numbers as these students will find out. In fact, the real numbers are a part of a larger system (and our next level of abstraction) of **metric spaces. **Students begin learning the structure of metric spaces and how metric spaces interact with one another via maps. Often these ideas are imparted to the students in a mathematical analysis course. Many of the ideas of metric spaces are still not restricted to this system. In fact, we have reached the next level of abstraction, **topology**. Here, students are no longer concerned with numbers and distances at all. Instead they simply have a notion of “closeness” in terms of open sets. They begin working with highly abstract spaces, often those that can’t even be visualized, yet the ideas of topology allow us to assess the deeper structure of mathematics. For example, manifolds helps us identify types of spaces that would, on the small scale, “look” the same as the some product of the real numbers. This might say something about the shape of the universe itself. For example, we could be sitting in a manifold right now, but it “looks” like 3-space on our very tiny scale. But there are problems in topology that are quite hard to solve. Namely, determining which spaces are homemorphic (can “morph” into) to each other. But there are many tools in abstract algebra that, if phrased in the right way, will be able to help. There must be a way to marry these two fields.

This is exactly what Saunders Mac Lane and Samuel Eilenberg achieved with

*General Theory of Natural Equivalences*, Transactions of the American Mathematical Society Vol. 58, No. 2 (Sep., 1945), pp. 231-294 (JSTOR).

Though it was not widely accepted (and in fact rejected), this article was published as seen above. Mac Lane went on to publish Category Theory for the Working Mathematician in 1971, which is widely accepted as THE introductory book for the subject. As described in an earlier post Categories and Functors, category theory looks at the most basic relationships between mathematical spaces. Many common patterns emerge as proofs in one specific setting are seen applied quite easily to the universal setting. This in effect can prove many important theorems at once. But is this as far as abstraction takes us? Can we go farther? The answer is most certainly yes. See higher category theory. There may be no end to our abstraction as subject. One day, it might even lead us to the very nature of Mathematics itself, and thus the nature of the language of the universe. Who knows what we’ll discover!

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Many of us have seen the complex numbers. These are numbers of the form where and are real numbers and . Perhaps more important to this story, a complex number can be interpreted as points in a plane. Instead of the usual Cartesian plane with and axes, we have a Complex plane with a real and imaginary axis. So, for example, we can plot the complex number as the point .

This plane has many interesting geometrical concepts. But it is still just a plane. So yet again, we’ve hit a wall. What if we wanted a number system that works nicely, but applies to our 3-space? Enter William Hamilton.

William Hamilton was an Irish mathematician and physicist concerned with this exact problem. In 1843 Hamilton, likely in an attempt to clear his mind, went for a stroll with his wife. While walking on that October day, they decided to cross Broome Bridge. But little did Hamilton know, mathematics inspiration waits for no one. It suddenly dawned on him!

William, beside himself, was absolutely not prepared for writing down such a discovery. So, he did what any self-respecting mathematician would do. Gone was the care for public property as he carved hard these rules into the stone of the bridge, his face grinning gleefully as he did so. Unfortunately, no trace of the carving remains today. Instead, there is just a plaque commemorating the event.

The Quaternion, as he called them, found their use in mechanics and other areas of Physics. Multiplication in this group can be interpreted as a rotation in 4-dimensional space. For more on the algebraic side of things, consult popular Abstract algebra textbooks such as Dummit and Foote.

Hamilton also introduced the Bi-quaternion. This is an 8-tuple with a multiplication rule. One might be led to ask the question, is there a 16-tuple that works nicely? How high can we go? There are a ton of methods for extending the real numbers. Check out the Cayley-Dickson construction and go from there. However, if we are talking about finite dimensional division algebras, we can only go to the Quaternion and no farther. This is Frobenius’ theorem. Hyper-complex numbers is an ever interesting field of mathematics. These not only extend the real numbers, but also extend our understanding of the inherent nature of that we call number.

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If you happen to have a ribbon or something like it close by, lay it out flat. Twist one end 180° and connect it to the other end. Now look at the object created. The Möbius strip is two dimensional. Notice it only has a length and width. That means that it is a two dimensional object that exists in three dimensional space! Not only that, if you attempt to press your strip down on a table, it starts to cross itself and lose its properties. So the strip cannot exist in 2D space.

Imagine, or try, to color one side of this strip. You may have realized that if you tried to color just one side, you actually color the whole thing! This means that this surface has only one side. That is, the Möbius strip is non-orientable. To see this further, fix the strip still in space. Now take a point and give it a direction “outwards.” If you move this point around the strip, by the time you return to its original position, the direction will be reversed! See the applet from Math Insights for an interactive visualization. Not only is this strip non-orientable, it is in some sense a basis for determining non-orientability of other surfaces as noted in the following theorem.

**Theorem: **A surface is non-orientable if and only if it contains a subspace that is homeomorphic to a Möbius strip.

On this last note, we leave a short exercise. Draw a line in the center of your Möbius band. Cut along that line and explore what happens.

Topology is full of fun and interesting shapes and surfaces. In the next post, we will look at some other topological spaces you can make with a simple rectangle! We will be exploring the strange and the unusual in this exciting series!

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The anthem of Topological Data Analysis is that data has shape and that shape matters. We would like to take a data sample and describe the topological space it was sampled from. This will help us make predictions to where new data may land. But what tools do we need? As the name suggests we are going to be looking in our topology textbooks. We need an invariant of homeomorphisms that can be described easily by linear algebra. This is so because we wish to be able to actually compute these things.

Homeomorphisms would be ideal, however they won’t work. It can be quite difficult to even come up with a homeomorphism. So we look to the next best thing, homotopy. Again homotopy groups are difficult to even write down, we wouldn’t want to try to code them. Okay, well there is one more ‘H’ word that might help us out, and you may have guessed it! It’s homology. Homology is invariant under homeomorphisms, meaning if two topological spaces are homeomorphic, then they have the same homology. Hence, if two spaces do not have the same homology, then they cannot be topologically the same. Also, homology is easily describable through linear algebra. This makes it incredibly easy to compute! The main problem is that two problems with the same homology may still not be homeomorphic. This problem is handled with the principle of Occam’s Razor and will be explained later.

In this post we will give an introduction to simplicial complexes. We will also define two simplicial complexes, the Rips and Čech complexes, that are quite popular in practice. So let us begin with the definition of a simplex.

**DEFINITION: **An **m****-simplex ** is the convex hull of *m*+1 points (called **vertices**) in . We describe a simplex by its vertices. i.e. will denote an *m*-simplex.

From left to right we have a 0-simplex, 1-simplex, 2-simplex, and 3-simplex.

If is an *m**–*simplex, and is an *n*-simplex for *n* < *m*, and if the vertex set of is contained in the vertex set of we say . We are immediately able to build structures with these simplices.

**DEFINITION: **A **simplicial complex ** is a collection of simplices satisfying the following two rules.

- If and then .
- If then or .

The first property is commonly known as *downward closure*. We refer to the second one as a *minimal incidence* property.

On the left is a simplicial complex. The structure on the right fails property 2.

**DEFINITION: **A **simplicial map** between, is a map so that whenever is a simplex in , we have is a simplex in .

**LEMMA: ** **Simp**, with objects as simplicial complexes and morphisms as simplicial maps, forms a category.

*Proof: *The reader is encouraged to try to prove this using the definition of categories in Categories and Functors.

While it is great to have an idea for what simplicial complexes look like, this definition is actually slightly too concrete. For this reason, we need to define abstract simplicial complexes. The idea of these objects is that they are abstract enough to be useful in theory and computation, and it is okay to worry about the nice space that they will fit in later.

**DEFINITION: **An **abstract simplicial complex ** is a finite collection of sets satisfying, and implies .

Notice we dropped the minimal incidence property as well as the necessity of having a vertex set of points in Euclidean space. The notion of a simplicial map is exactly the same. Hence we obtain a category **AbSimp**. What we will see next is a justification for using abstract simplicial complexes. First note, that for every simplicial complex there is an abstract simplicial complex with the same vertex set of .

**LEMMA: **There is a functor so that for all and for all .

This lemma simply says that we can associate each abstract simplicial complex with a simplicial complex in a nice way. This allows us to work with these abstract objects and then fit them into a nice space later.

Now, we will take a look at two popular abstract simplicial complexes. One gives an accurate description of the space but is not easily computable, while the other is not as accurate but is easily computable.

**DEFINITION: **Let be a finite set of points in some metric space. Let be a positive real number. We define the **Čech complex at scale **to be the set of all simplices whose vertices lie in and the intersection of the balls centered at these vertices with radius is nonempty. In symbols

Where denotes the ball of radius centered at .

The Nerve Theorem tells us that our space is properly described by this complex. However, computationally, this complex is taxing. The problem is one is trying to find the intersection of metric balls which is much harder than just simply checking a condition. So we look to the next complex.

**DEFINITION:** Let be a finite set of points in some metric space. Let be a positive real number. We define the **Rips complex at scale **to be the set of all simplices whose vertices are within of each other. In symbols,

Notice the Rips condition is simply one we have to check. Though as previously stated it does not give an accurate description of the space since as we see in the picture we would fill the triangle in well befor the Čech complex would. However! It does it “well enough” as described by the following Lemma.

**LEMMA: (RIPS) ** .

Really this thing says that the Rips complex approximates the Čech complex well enough.

That will do it for part 1. It is meant to be an introduction to simplicial complex. Part 2 will cover enough Homology to understand Persistence. In the future we will look at concrete examples and the categorification of the field.

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The anthem of Topological Data Analysis (TDA) is that data has shape and that shape matters. We would like to take a data sample and describe the topological space it was sampled from. This will help us make predictions to where new data may land. TDA has been used in many fields such as medical imaging [1] , sensor networks [2], sports analysis [3], disease progression [4], image analysis [5], signal analysis [6], and many others.In this post, we are just going to give the basic idea. Suppose we have are given a data set that looks like this.

It seems obvious to the human eye that this data has been sampled from circular object. This is because we are wired to recognize patterns, especially ones as easy as this data set. But how could we get a computer to understand this pattern? This is where TDA comes in. Imagine that we begin growing balls around points.

As the balls grow they will intersect. When two balls intersect, we place a line segment (edge). When three balls intersect we place a triangle. When four balls intersect we place a tetrahedron and so on.

Eventually, the balls will have grown enough to bound a gap.

As we continue growing the balls, the gap will eventually close. Beyond this point nothing changes topologically, hence we can tell the computer to stop here. Now what we have done is created what is called a **filtration** which is simply an increasing chain of spaces. To capture the topological properties, we use homology to count holes. We apply homology (count the holes) to each space in the filtration. Then, more or less, we measure how long the holes last. The idea is that the longer lasting holes are more important to the topological properties of the space the data was sampled from. This process is accurately called **persistent homology**. There are, of course, some fine details excluded from this summary, especially the fact that TDA does not begin nor stop at persistent homology. If you would like to know more please check out some of the references I am leaving at the bottom. I will be making a post (or series of posts) soon that will go a little deeper in the theory.

REFERENCES

The first 6 references are applications of persistent homology.

[1] Lee, Hyekyoung, et al. “Persistent brain network homology from the perspective of dendrogram.” *Medical Imaging, IEEE Transactions on* 31.12 (2012): 2267-2277.

[2] De Silva, Vin, and Robert Ghrist. “Homological sensor networks.” *Notices of the American mathematical society* 54.1 (2007).

[3] Goldfarb, Daniel. “An Application of Topological Data Analysis to Hockey Analytics.” *arXiv preprint arXiv:1409.7635* (2014).

[4] Nicolau, Monica; Levine, Arnold J.; Carlsson, Gunnar (2011-04-26). “Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival”. Proceedings of the National Academy of Sciences 108 (17): 7265–7270.

[5] Bendich, P.; Edelsbrunner, H.; Kerber, M. (2010-11-01). “Computing Robustness and Persistence for Images”. *IEEE Transactions on Visualization and Computer Graphics* **16**(6): 1251–1260.

[6] Perea, Jose A.; Harer, John (2014-05-29). “Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis”. Foundations of Computational Mathematics 15 (3): 799–838.

The next few are just references for one who would like to get started in studying the subject.

[7] Edelsbrunner, Herbert, and John Harer. *Computational topology: an introduction*. American Mathematical Soc., 2010.

[8] Bubenik, Peter, and Jonathan A. Scott. “Categorification of persistent homology.” *Discrete & Computational Geometry* 51.3 (2014): 600-627.

[9] Lesnick, Michael. “The theory of the interleaving distance on multidimensional persistence modules.” *Foundations of Computational Mathematics* 15.3 (2015): 613-650.

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Our undefined terms will be **set** and the binary relation of **membership**, when we write it will read as “ is a member of . We will use capital letters and to represent sets.

**(1) AXIOM (of Extensionality) : **For every and for every and for every if implies and is implies then . This can also be stated as

.

We have here what it means for two sets to be equal. That is, two sets are equal if they contain the same members. The next axiom guarantees the existence of subsets in a way.

**(2) AXIOM (Schema of Comprehension): **Suppose is a property of z. For every there exists so that if and only if and is true. We see this as,

**DEFINITION: **The set is a subset of the set if and only if implies , in this case we write .

Suppose we are given . Then for any we are guaranteed, by the Axiom Schema of Comprehension, that there is some whose elements all satisfy . But since there is nothing that satisfies we conclude contains no elements. Notice, as of right now, our depends on a given . Suppose there is some other set, with no members. Then the statement is vacuously true as the antecedent is always false. We use the same reasoning to see that the statement . We conclude then by the Axiom of Extensionality that . (this is actually a proof of the proposition: “There exists a unique set with no elements”) Hence, the set with no elements exists and is unique, so we obtain the following definition.

**DEFINITION: **The set with no elements will be called the **empty set** and will be denoted, .

Notice the empty set is a subset of every set since the statement is vacuously true. The next axiom allows us to “build” sets in a way. It tells us that we can find a set containing two given sets.

**(3) AXIOM (of Pairing): **For any and , there exists so that and .

Notice that this axiom does not give us that contains only and . For this we have the following proposition.

**PROPOSITION: **For any pair there exists a unique set containing exactly and as its only members.

*Proof: *Let be a set containing and by the Axiom of Pairing. Let be the statement “ or “. By logic and the Axiom Schema of Comprehension we find that exists. We conclude that is unique by the Axiom of Extensionality.

From the Axiom of pairing we also get for any set the set is valid. And, we get that . This will come in handy later but first,

**(4) AXIOM (of Union): **For any set there exists a set whose elements contain all of those elements in the members of .

Let and be sets. We can use the Axiom of Pairing to obtain a set . Then the axiom of union guarantees exists. But what about intersections? We have actually had the definition of intersection all along. Let be the statement ““. Then we obtain by the Axiom Schema of Comprehension, there is a containing elements of for which is true. But this makes exactly the set of all for which and . Hence . We obtain the following definitions.

**DEFINITION: **Let and be sets. The **union** of and is the set where if and only if or $latex z\in y.

The **intersection** of and is the set for which if and only if and .

**(5) AXIOM (of Replacement): ** Let be a property such that for every there is a unique so that holds. Then for every there exists a such that for every there is a so that is true.

We can use this axiom along with the Axiom Schema of Comprehension to show that there is a set containing exactly the that satisfy . We can show it is unique by the Axiom of Extensionality. (Try this!)

**(6) AXIOM (of Infinity): **There is a set so that and whenever , .

From this we find , , and , and so on. Let be the property that is one of these sets. Then we define the set to be the set of all so that is true. This is no mistake for this set is exactly what we will call the natural numbers. In this way we replace these sets with notation, There are many more intricacies of this process (such as uniqueness) that we will not delve into in this post. For our subject is that of Zermelo-Fraenkel axioms. So we shall proceed from this topic.

**(7) AXIOM (of Power Set): **For every set there exists a set so that implies .

From this axiom (and the others we have used) we can define the unique power set of a set (prove it!).

**DEFINITION: **Let be a set. The **power set** of , , is the set which contains exactly all of the subsets of .

The next axiom will complete the list for Zermelo-Fraenkel (ZF) set theory.

**(8) AXIOM (of Foundation): **Every nonempty set (a set so that ) contains a member so that .

.

The Axiom of Foundation tells us that if is a set then . (Why?) Then we see that in this system Russell’s Paradox is not possible. The final axiom is the Axiom of Choice. This is a very controversial axiom, so in the interest of space we will simply state it in this post.

**(9) AXIOM (of Choice): **Suppose is a collection of nonempty sets. There exists a function that assigns to each set an element .

The axiomatic system formed by (1)-(9) is known as ZF+C or ZFC and is the most widely used and accepted system in mathematics today. In the next post we will explore the Axiom of Choice and its equivalents. We have seen that, by assuming the Axiom of Choice, it is possible to split a sphere into two spheres that are congruent to the original. The next axiomatic system we will look at is Bernays-Gödel-von Neumann axiomatics as presented in Dugundji’s* **Topology. *This will be the last set theory system we will explore for a while.

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It isn’t too often that we think about or even explore the fundamental building blocks of mathematics. These building blocks are called **axioms. **Axioms are statements taken to be true, i.e. they cannot be proven. This causes mathematicians, whether they know it or not, to take a lot of things on faith. All of modern mathematics is built on top of **axiomatic systems**, which is a list of axioms from which one can logically derive theorems. We further require that no axiom on the list can be proven by using other axioms. In this series we will look at some interesting axioms and axiomatic systems that govern mathematics. We will begin with perhaps one of the most important systems and that is sets. One could argue that the system of logic used is the most important. We are assuming a working knowledge of tools used in logic, and will forgo formalizing such a thing. We often think about an axiomatic system like a tree.

But first let us get a feel for why an axiomatic system is so important. This brings us to something called **naive set theory**. This naive set theory is an example of common occurrences in mathematics in which we define things by using natural language. We often wind up tossing words around until someone formalizes an idea. This happened with set theory.

**DEFINITION: **The collection of all things satisfying a property forms a **set**. In symbols, if is the statement “ has property “, then a set can be written in the form .

From this definition one can deduce that the collection of all things that are sets is a set. In symbols, if is the statement is a set, then is a set. But this means contains itself. Now that we know one exists, let us call all sets that contain themselves **abnormal**. In symbols, a set is abnormal if . Where means is an element of.

Obviously does not contain itself as a set. We will call these sets **normal**. In symbols, a set is normal if . It is clear that these two definitions are exclusionary and all sets are either normal or abnormal. Now consider the following set.

.

This is simply the collection of all normal sets. By definition, this is indeed a set. But now it must either be normal or abnormal. If is normal, then . By definition of we have which is a contradiction.

On the other hand if is abnormal then , hence by definition of we have $R\notin R$ another contradiction. We conclude that is neither normal nor abnormal. Which is our ultimate contradiction which tells us that naive set theory is inconsistent.

This example is known as Russell’s Paradox (1901) and it has many equivalents in naive set theory). Even though naive set theory has this major flaw, it is still very useful in teaching and introducing students to the real heart of mathematics. In the next post we will explore the Zermelo-Fraenkel (ZF) axiomatic system which is the most widely used system in mathematics today. We will also make note of ZFC which includes the historically controversial Axiom of Choice.

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