# Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

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1 Before we start, I want to point out that these notes are not checked for typos. There are prbally many typeos in them and if you find any, please let me know as it s extremely difficult to find them all - by myself. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

2 So what is set theory anyways? Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

3 So what is set theory anyways? A better question is What is mathematics anyways?. Seriously! What is it? We understand what it means to add to get 3, but this process exists only in thought. What, even, does the concept of 2 mean? Do we assume it already exists or does it exist because of some other reason? Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

4 So what is set theory anyways? A better question is What is mathematics anyways?. Seriously! What is it? We understand what it means to add to get 3, but this process exists only in thought. What, even, does the concept of 2 mean? Do we assume it already exists or does it exist because of some other reason? The more advanced we get mathematically, e.g., algebra, calculus, geometry, etc we use concepts like these over and over without ever thinking about where they originated. Is it possible that what we think makes perfect sense to us actually contradicts something else that also makes perfect sense? Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

5 So what is set theory anyways? A better question is What is mathematics anyways?. Seriously! What is it? We understand what it means to add to get 3, but this process exists only in thought. What, even, does the concept of 2 mean? Do we assume it already exists or does it exist because of some other reason? The more advanced we get mathematically, e.g., algebra, calculus, geometry, etc we use concepts like these over and over without ever thinking about where they originated. Is it possible that what we think makes perfect sense to us actually contradicts something else that also makes perfect sense? To answer the question what is set theory? here is an analogy: If modern mathematics is like building a city, then set theory is the very beginning of that city s construction starting with the atoms from which that city is created. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

6 Axioms To build anything, you have to start with something. To build mathematics, we start with the axioms of set theory. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

7 Axioms To build anything, you have to start with something. To build mathematics, we start with the axioms of set theory. Definition (Axiom) An axiom is a starting point of reasoning from which other statements are logically derived. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

8 Axioms To build anything, you have to start with something. To build mathematics, we start with the axioms of set theory. Definition (Axiom) An axiom is a starting point of reasoning from which other statements are logically derived. Unlike theorems, axioms cannot be demonstrated by proof. If they could, they would be called theorems. Axioms are the basic things we accept to be true. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

9 Axioms To build anything, you have to start with something. To build mathematics, we start with the axioms of set theory. Definition (Axiom) An axiom is a starting point of reasoning from which other statements are logically derived. Unlike theorems, axioms cannot be demonstrated by proof. If they could, they would be called theorems. Axioms are the basic things we accept to be true. What we decide to be an axiom is up to us, but classically, they are chosen to be the things that are so self evident that they are accepted without controversy. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

10 The Axioms of Zermelo-Fraenkel Set Theory (ZFC) A widely accepted framework to build modern mathematics are the axioms of Zermelo and Fraenkel (ZF). In 1908 Zermelo introduced the axioms and in 1922 Fraenkel added the additional axiom of replacement for which Zermelo later admitted to forgetting to include in his work. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

11 The Axioms of Zermelo-Fraenkel Set Theory (ZFC) A widely accepted framework to build modern mathematics are the axioms of Zermelo and Fraenkel (ZF). In 1908 Zermelo introduced the axioms and in 1922 Fraenkel added the additional axiom of replacement for which Zermelo later admitted to forgetting to include in his work. The axiom of choice (AC) is one of the most controversial axioms of set theory. In this class, we will develop as much as possible only using ZF, but at some point we will assume AC. ZF with AC is called Zermelo-Fraenkel with choice (ZFC). We will see why AC is controversial and I will leave it up to you to decide on if you accept it or not. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

12 Sets Sets are a collection of objects. These objects are called elements of the set. NOTE: A set is not the objects themselves but the collection as a whole. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

13 Sets Sets are a collection of objects. These objects are called elements of the set. NOTE: A set is not the objects themselves but the collection as a whole. One of the defining characteristics of ZF is that everything is a set, that is, sets will contain objects, but these objects are also sets, etc. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

14 Sets Sets are a collection of objects. These objects are called elements of the set. NOTE: A set is not the objects themselves but the collection as a whole. One of the defining characteristics of ZF is that everything is a set, that is, sets will contain objects, but these objects are also sets, etc. Sets don t have a formal definition - they exist because the axioms of ZF say they exist. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

15 Sets Sets are a collection of objects. These objects are called elements of the set. NOTE: A set is not the objects themselves but the collection as a whole. One of the defining characteristics of ZF is that everything is a set, that is, sets will contain objects, but these objects are also sets, etc. Sets don t have a formal definition - they exist because the axioms of ZF say they exist. On the other hand, to give a definition, we need to make sure that the set that is being defined exists, that is, we cannot create sets from definitions. In other words, definitions give names to things that are already there. In addition, we do not want to give a single name to multiple things, that is, we need to make sure our definitions are well defined. This will come up over and over again. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

16 ZF Axiom (1. The Axiom Of Existence) There exists a set which has no elements. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

17 ZF Axiom (1. The Axiom Of Existence) There exists a set which has no elements. This is the only axiom we will have where we claim something out of thin air exists. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

18 ZF Axiom (1. The Axiom Of Existence) There exists a set which has no elements. This is the only axiom we will have where we claim something out of thin air exists. In logical symbols this axiom can be written as Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

19 ZF Axiom (1. The Axiom Of Existence) There exists a set which has no elements. This is the only axiom we will have where we claim something out of thin air exists. In logical symbols this axiom can be written as x y(y / x) Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

20 ZF Axiom (1. The Axiom Of Existence) There exists a set which has no elements. This is the only axiom we will have where we claim something out of thin air exists. In logical symbols this axiom can be written as x y(y / x) You might ask Why do we assume there exists a set with no elements? Why not just assume there exists a set. Well, we could have done that and with a later axiom we could prove that there exists a set that has no elements. However, from a minimalist point of view, the set with no elements seems the best to start with. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

21 ZF First question: Is there more than one set that contains no elements? Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

22 ZF First question: Is there more than one set that contains no elements? At this point, we can t answer this question we need another axiom Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

23 ZF Axiom (2. The Axiom of Extension) If two sets X and Y have the same elements, then X = Y. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

24 ZF Axiom (2. The Axiom of Extension) If two sets X and Y have the same elements, then X = Y. The more common way of saying this is: If every element of X is an element of Y, and every element of Y is an element of X then X = Y. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

25 ZF Axiom (2. The Axiom of Extension) If two sets X and Y have the same elements, then X = Y. The more common way of saying this is: If every element of X is an element of Y, and every element of Y is an element of X then X = Y. You might ask why even bother saying this as it sounds so obvious. Well, up until this point, we did not have a definition of what = means. Now we do. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

26 ZF Axiom (2. The Axiom of Extension) If two sets X and Y have the same elements, then X = Y. The more common way of saying this is: If every element of X is an element of Y, and every element of Y is an element of X then X = Y. You might ask why even bother saying this as it sounds so obvious. Well, up until this point, we did not have a definition of what = means. Now we do. In symbols this is written as X Y ( y Y (y X ) x X (x Y ) X = Y ) Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

27 ZF We can now answer the previous question: There is, in fact, only one set that contains no elements Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

28 ZF We can now answer the previous question: There is, in fact, only one set that contains no elements Proof. Suppose there are two sets X and Y that have no elements. It is a true statement to say: x X x Y. because x X is a false statement (see truth tables for ). It is also the case that y Y y X for the same reason. By the axiom of extension X = Y and therefore there is only set that contains no elements. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

29 Therefore, by the axiom of extension and the axiom of existence we can now make a definition: Definition (The empty set) The set that contains no elements will be called the empty set and denoted by the symbol or sometimes by {}. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

30 Therefore, by the axiom of extension and the axiom of existence we can now make a definition: Definition (The empty set) The set that contains no elements will be called the empty set and denoted by the symbol or sometimes by {}. Notice this definition is giving a name to the set that has no elements. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

31 Therefore, by the axiom of extension and the axiom of existence we can now make a definition: Definition (The empty set) The set that contains no elements will be called the empty set and denoted by the symbol or sometimes by {}. Notice this definition is giving a name to the set that has no elements. Until now, it was not logical to make the above definition. We had no yet answered the question of is there more than one set that contains no elements? We don t want to call multiple things by a single name. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

32 The axiom of extension also tells us about repetition of objects in a single set: Example Prove that the set {A, A} = {A}. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

33 The axiom of extension also tells us about repetition of objects in a single set: Example Prove that the set {A, A} = {A}. Proof. Consider the statement x {A, A}. The only thing x can be is A and since A {A} it is true to say x {A, A}(x {A}). Similarly, x {A}, (x {A, A}). So by the axiom of extension {A, A} = {A}. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

34 Russell s Paradox So we know the empty set exists and we know how to tell when two sets are equal. How do we get our hands on more sets? Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

35 Russell s Paradox So we know the empty set exists and we know how to tell when two sets are equal. How do we get our hands on more sets? Intuition tells us that the elements of a set should be defined by a property those elements should have. For a given property P(x) that an arbitrary element x should have, we should get a set S that contains all elements x that have property P(x). Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

36 Russell s Paradox So we know the empty set exists and we know how to tell when two sets are equal. How do we get our hands on more sets? Intuition tells us that the elements of a set should be defined by a property those elements should have. For a given property P(x) that an arbitrary element x should have, we should get a set S that contains all elements x that have property P(x). In other words, we want to say is a set! S := {x P(x)} Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

37 Russell s Paradox So we know the empty set exists and we know how to tell when two sets are equal. How do we get our hands on more sets? Intuition tells us that the elements of a set should be defined by a property those elements should have. For a given property P(x) that an arbitrary element x should have, we should get a set S that contains all elements x that have property P(x). In other words, we want to say is a set! S := {x P(x)} There is a subtle problem with this idea. It s called Russell s Paradox Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

38 Russell s Paradox In 1901, Bertrand Russell pointed out the following paradox by defining a set R = {x x / x} Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

39 Russell s Paradox In 1901, Bertrand Russell pointed out the following paradox by defining a set R = {x x / x} Notice here that the property P(x) is that the set x is not an element of itself. I agree, that is a strange property, but still, it s a property. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

40 Russell s Paradox In 1901, Bertrand Russell pointed out the following paradox by defining a set R = {x x / x} Notice here that the property P(x) is that the set x is not an element of itself. I agree, that is a strange property, but still, it s a property. Now let s ask is R R? Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

41 Russell s Paradox In 1901, Bertrand Russell pointed out the following paradox by defining a set R = {x x / x} Notice here that the property P(x) is that the set x is not an element of itself. I agree, that is a strange property, but still, it s a property. Now let s ask is R R? Well if R R then R must have the property that R / R so that can t be. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

42 Russell s Paradox In 1901, Bertrand Russell pointed out the following paradox by defining a set R = {x x / x} Notice here that the property P(x) is that the set x is not an element of itself. I agree, that is a strange property, but still, it s a property. Now let s ask is R R? Well if R R then R must have the property that R / R so that can t be. The other option is that R / R. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

43 Russell s Paradox In 1901, Bertrand Russell pointed out the following paradox by defining a set R = {x x / x} Notice here that the property P(x) is that the set x is not an element of itself. I agree, that is a strange property, but still, it s a property. Now let s ask is R R? Well if R R then R must have the property that R / R so that can t be. The other option is that R / R. If R / R, then R is a set that has the property that it s not an element of itself. But then R R by definition of R so that can t be either. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

44 ZF So no matter what way we look at this set there is a problem with it s definition. This problem is solved in ZF set by the following: Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

45 ZF So no matter what way we look at this set there is a problem with it s definition. This problem is solved in ZF set by the following: Axiom (3. The Axiom of Comprhension) Let P(x) be a property of x. For every set A, there is a set S such that x is an element of S if and only if x A and P(x). Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

46 ZF So no matter what way we look at this set there is a problem with it s definition. This problem is solved in ZF set by the following: Axiom (3. The Axiom of Comprhension) Let P(x) be a property of x. For every set A, there is a set S such that x is an element of S if and only if x A and P(x). Secretly, this axiom tells us that subsets are also sets. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

47 ZF So no matter what way we look at this set there is a problem with it s definition. This problem is solved in ZF set by the following: Axiom (3. The Axiom of Comprhension) Let P(x) be a property of x. For every set A, there is a set S such that x is an element of S if and only if x A and P(x). Secretly, this axiom tells us that subsets are also sets. This also tells us how to define sets by properties, i.e., the axiom of comprehension says for any property P(x) and any set A is a set! S := {x A P(x)} Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

48 In symbols the axiom of comprehension can be written as A S x (x S x A P(x)) Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

49 In symbols the axiom of comprehension can be written as A S x (x S x A P(x)) The important statement to take away from the above is S. The axiom of comprehension tells us that if we define things in this way, S is a set. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

50 In symbols the axiom of comprehension can be written as A S x (x S x A P(x)) The important statement to take away from the above is S. The axiom of comprehension tells us that if we define things in this way, S is a set. Remember, the point of these axioms is to give precise instructions on what is and what is not a set. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

51 In symbols the axiom of comprehension can be written as A S x (x S x A P(x)) The important statement to take away from the above is S. The axiom of comprehension tells us that if we define things in this way, S is a set. Remember, the point of these axioms is to give precise instructions on what is and what is not a set. Now notice Russell s Paradox is avoided by requiring, first A to be a set, and then setting S A. Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15

### Now, with all the definitions we ve made, we re ready see where all the math stuff we took for granted, like numbers, come from.

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