Introduction to Logic and Axiomatic Set Theory

Transcription

1 Introduction to Logic and Axiomatic Set Theory 1 Introduction In mathematics, we seek absolute rigor in our arguments, and a solid foundation for all of the structures we consider. Here, we will see some basic ideas about how this happens. We will begin with a discussion on mathematical logic. This post has a bit of fairly technical exposition and not a lot of exercises. We try hard to make the post more accessible, but as you will see, this required omission of many interesting details, and too much omission would remove the point of the post. The reader should feel free to skip or skim some sections, as the difficulty level of the material varies throughout. Here, we will start with an exploration of the nature of mathematical languages and statements, and what it means to prove such statements. Once we have a solid way of writing formal mathematical arguments, we want to start constructing mathematical structures to study. The most ubiquitous way to formalize mathematical structure is with what is known as Zermalo-Fraenkel set theory together with the Axiom of Choice (abbreviated as ZFC). ZFC is a list of axioms which define the objects we will call sets, and the operations we may do with sets. Later, we will define ZFC and discuss some basic implications. From there, in theory all mathematical objects can be defined as a set in the universe defined by ZFC. More importantly, we hope that the universe is robust enough to allows to prove many interesting theorems, but will not let us prove anything contradictory. We will discuss some fundamental issues in axiomatic logic based on this goal. Structures that we study range from very simple objects like finite sets of objects (for example, the set {1, 2, 3}), up to much more complicated structures like differentiable manifolds and algebraic varieties. In formal mathematics, we build structures out of sets. Through basic set theory operations, we can essentially build up arbitrarily complicated objects. In practice, during modern research outside of set theory and logic, mathematicians do not usually focus on the set-theoretic basis for the objects they study, but it is important for every mathematician to understand the foundational ideas behind their work. In this post, we will be avoiding some of the foundational and philosophical questions and problems which arise in this study. Pursuing those issues would be better fit for a several-hundred-page treatise on the philosophy of mathematics. This includes ideas on the nature of truth in mathemat- 1

2 ics and what it means for a statement to mean something, among other things. We will also avoid the problem that in order to define a set, we need a language to work in, but in order to define a language, we need a set of characters and syntax rules. We also will be avoiding a lot of theorems on the validity of these approaches, as they cover ideas which are beyond the scope of this post. Hopefully the reader can excuse these omissions, and we do our best to present the topics in a logical and meaningful manner. Section 2 will be based on a couple early sections of the illustrious Yuri Manin s book, A Course in Mathematical Logic for Mathematicians, and some of section 3 will draw from a couple early sections David Marker s book, Model Theory: An Introduction. 2 Mathematical Languages Every good statement in mathematics has an unambiguous meaning in its universe describing some kind of relation between objects. Most of the time, this is written out in words, like in proofs. But to be fully rigorous, we always need to be sure that the statements in a proof can be translated into a formal mathematical language which allows for this unambiguous interpretation. If you study mathematical logic and set theory, you will find that everything gets a very specific definition. We start with the definition of a language. First, we require an alphabet, which is any collection A from which expressions are made. The language itself will be a distinguished collection of expressions to which we may assign a meaning. We will explore the abstract notion of a language by defining the firstorder language of ZFC (denoted L 1 Set) as our primary motivating example. First-order means that we may not quantify over arbitrary properties or subsets of the universe, and we may only quantify over objects in the universe. What this means exactly will be more clear later. We will not be exploring higher-order languages in this post. The alphabet A of a formal language is made up of six disjoint subsets, each of which has characters used for a specific purpose. We will define each of these and give examples from L 1 Set. Connectives and Quantifiers This consists of the logical symbols we learned in the course, such as,,,,,, and. In a first-order language, we may only use and to quantify over the whole universe of objects. Variables These are just letters we use to denote arbitrary objects, such 2

3 as x and y. Constants These are used to denote fundamental objects from which we build other objects. In L 1 Set, the only constant is. Operations of some degree These are special functions which are fundamental to the axiomatic system which we can perform on the elements. There are none in L 1 Set, but in Peano Arithmetic they are S, + and. The degree refers to the number of arguments; both + and are degree two and S is degree 1. Relations of some degree These tell us how objects are related to each other. In L 1 Set, there is (inclusion) and = (equality). Parentheses We actually need to make formal the use of parentheses, since we wish to use them in our logical statements! Now, an expression in the language may be either a term or a formula. Terms are either variables or constants (these are atomic terms, since all other terms are made of them), or if f is an operation of degree r and t 1,..., t r are terms, then f(t 1,..., t r ) is a term. We can think of terms as specifying elements in the universe. Formulas are similar. Atomic formulas are of the form p(t 1,..., t r ) for relation p of degree r and terms t 1,..., t r, and nonatomic formulas are formulas together with a logical connective or quantifier. For example, if P and Q are formulas and x is a variable, then the following are nonatomic formulas: P Q, P Q, P Q, P Q, (P ), x(p ), x(p ). Here is an example of a formula in L 1 Set together with an intuitive meaning. y(y z y x). This is the precise definition that z is a subset of x. Once we have defined this, we might use the common shorthand z x, but the symbol is not part of our formal language. So, if you ever see the symbol in a proof, it is already using shorthand beyond the basic language of set theory! Imagine if you had to write all of your proofs using the formal L 1 Set - the fact that this is extremely impractical for Math 0 material should hint at how important shorthand notation is for all of mathematics. If all terms in a formula are constants (i.e. there are no variables), we can think of it as a statement which may be assigned a truth value, and 3

4 hence may or may not be true in some universe. If there are variables, we might think of the formula as specifying the collection of terms which, when substituted for the variable, would make the statement true. This notion will be explored shortly. Now we have a language, and we can write all statements we would want in it. We will end this section with a few additional examples and exercises on translation between formal L 1 Set and our intuitive colloquial way of discussing of the structures involved. Example: z(z x z y) x = y This can be taken to be a definition of equality in set theory. Two sets x and y are equal if and only if all elements in one are also in the other. Otherwise stated, a set is precisely determined by its elements. Exercise: How would you write this out in set theory notation? The empty set is empty. Another more formal way to say this is, for all objects it is false that the object is an element of. Exercise: How would you write this out in English? u v x z(z x (z = u z = v)) Hint: given any u and v, this says that there is a special set x that is built from u and v in a certain (simple) way. This is also known as the Axiom of Pairing, and is important in ZFC. Now, we have our set of symbols and rules (syntax) for them, and we know how to translate between the formal language and the intuitive way of understanding our objects. We next need to understand how to use the statements which allow us to construct a universe in which we will actually do mathematics. 3 Mathematical Statements and Axioms To build a universe, we first need to understand what mathematical statements really are. We will consider the two main types of statements in first-order logic: propositions and predicates. Propositions are simple statements which may be assigned a truth value (true or false). 2 is a prime is a proposition, as is all differentiable functions are continuous or 5 is an even number. These have no variables, and in a suitable language, could be written out with constants as a formula. The second basic type of statement is known as a predicate. Predicates take variables, and when the variables are filled with constants, become propositions. Some examples might be, x is an odd number, or x + y + z = 1. We can also think of predicates as functions with codomain {True, 4

5 False}. Again, in a suitable language, we could write these as formulas with variables. It is also important to have at least a brief idea of what a proof really is. A technical definition of a proof might be as follows. Definition A proof of some result is a finite sequence of statements, such that each statement follows from a previous statement, axiom, or theorem, and the last statement is the desired result. This makes sense because we are essentially turning previous results into the new theorem. Our axioms (and prior theorems) allow us to prove new results. All that remains to understand is our notion of follows from. This relies on our logical laws of inference. Logical laws of inference are essentially a collection of rules which allow us to transform formulas with logical connectives into new formulas. They can be stated as a list of pairs of statements such that we may turn the first statement into the second statement. One easy example is, P Q, therefore P. This means that we may take the statement P Q as one line in a proof, and the statement P may be a later line, since if both P and Q are true, then of course P is true. Another rule is, P (Q Q), therefore P. This rule is our basis for proof by contradiction. It means that, if a statement P implies a contradiction (Q Q) then P must be false (that is, we know P holds). There are many other rules which are of a similar flavor. Essentially, they are formalizations of our intuitive way of interpreting logical connectives, and they serve as an additional collection of rules behind the scenes of formal languages. This allows us to in theory write proofs as statements in entirely formal language. In practice of course, we also include regular language to explain our reasoning and make sure the reader is able to follow. It is important to keep in mind that every step of a proof should be possible to write in formal language. Each sentence of a proof should either be in formal language, a direct translation of the formal language, or an explanation of a statement in formal language. Thinking like this is not prevalent in everyday mathematical work outside of logic, but it is good practice to attempt it when first learning proofs, and to keep it in the back of one s mind when working in more complicated settings. Next, we want to actually build our universe. Let L be a language, and let S be some set of sentences (statements or propositions) in L. We say S is a L-theory. This will essentially be our set of axioms for some type of object; e.g. the theory of groups, the theory of sets, the theory of topological 5

6 spaces, and so on. Now, a model M of S essentially consists of a universe U, which is (sort of) a set, together with functions and relations for each function and relation in L, such that all sentences in S are satisfied. The universe will be the underlying set for some object. For example, the Peano axioms in chapter 7 might be a theory in the language of Peano arithmetic, and N with the usual operations is a model of this theory. Some theories may admit many different models. For example, if we remove the Peano axiom stating that 0 is not a successor to any number, then we could have {0, 1} with S(0) = 1 and S(1) = 0 as a model of the modified collection of axioms, in addition to the original N (check this). Some theories may admit no models. An easy example of an unsatisfiable theory in the language of Peano arithmetic would be the theory consisting of the sentences x : x 0 and x : x = 0. But these are contradictory, so no model can satisfy both sentences. The study of theories and models comprises an entire field of study aptly named Model Theory. Model theory fills up entire textbooks, and gets rather involved pretty quickly, so we will end our discussion of models here. At this point, we will take a slight change of direction and think about perhaps the most important mathematical universe: Zermalo-Fraenkel Set Theory. 4 Mathematical Structures as Sets We have seen how to define mathematical structures in a totally abstract way using model theory. But we often like to think of most mathematical objects as sets, even if they aren t explicitly defined as sets. This lets us have a very rigorous and concrete formulation for what would otherwise be very fuzzy structures. In this effort, the early 20th century saw the development of ZFC, which is intended to be a universal set theory which in some sense includes all meaningful mathematical objects. First, we must be sure that this is even a good goal - after all, many mathematical objects we have seen are not defined as sets! As it turns out, virtually all mathematical structures can actually be defined explicitly or implicitly as sets. For example, a function f : A B can be defined as a set of ordered pairs: f = {(a, b) a A, b B, f(a) = b}. An ordered pair can even be defined as a set: (a, b) = {{a}, {a, b}}. Ordered pairs are just a set of two objects with one distinguished as first object. We even define 6

7 natural numbers as sets, recursively: if n = 0 n N : n = { } if n = 1 n 1 {n 1} if n 2 Exercise: Write down the number n explicitly as a set. Prove this using induction. Exercise: Prove that (a, b) = (c, d) as sets if and only if a = c and b = d. You might also want to think about how we can define integers and rational numbers as sets. There are many ways, some of which are more commonly used than others. Real numbers are a bit more complicated to build from sets, but it is possible. Constructing the real numbers would be worthy of its own post! However, once we get beyond some very basic structures, it often becomes more convenient and meaningful to construct more complicated structures in terms of the simpler ones. This is similar to using the shorthand which we defined earlier. Now that we have seen that many mathematical objects can be realized as sets, we need to think more carefully about what a set actually is. 5 What is a Set? In the first paragraph of the chapter notes, it is mentioned that formal sets are quite complicated! Shedding some light on this is one purpose of this post. We intuitively want to define a set as any collection of objects, but that doesn t quite work. That definition of set is the basis for what is known as naive set theory. Why can t we just use this definition? One answer is that it leads to paradoxes that give impossible sets. By this naive definition, the following is a set: A = {X X / X} for all sets X. This is known as Russell s Paradox, named after Bertrand Russell. Think about why this is a paradox. In order to avoid problems like this, we can construct rules to define exactly what is and is not a set. First, we will declare that all mathematical objects must be sets. What this means is, there are no other basic objects: everything is a set, and is the only fundamental (constant) object. We 7

8 can put other names on the objects in the universe, like number or function, but each is just a set of a particular form. Next, we may think of the notion of containment as having no intrinsic meaning; it is just a relation that some sets have. This means that the symbol is a binary relation between sets that must be true or false for every pair of sets. We can intuitively think of a b as meaning that a is inside b, but it is really just that the relation is true for a and b (in that order). The important thing is just that now, our universe consists of all sets and this binary relation. For the purpose of discussion, we often still use language that fits the intuitive definition, but we really just mean this relation. 6 Zermalo-Fraenkel Axioms Now, we build our universe from scratch. We don t have any actual rules yet - so far we have just cleaned up our universe of objects and said very loosely what we want. The next step is to write down a list of rules that tell us how to build sets out of each other. We want to use only the basic logical language of L 1 Set. The Wikipedia page for ZFC has a list of axioms, so I will reproduce only a few here to demonstrate. It is not a very useful exercise to just list and describe all the axioms, so we will just discuss a few. Remember that the idea here is to define rules that a universe (model) for ZFC must satisfy. They state what sets will exist in terms of more basic sets (all the way down to ). As a very useful, interesting, and fun exercise, you should think about each of these, and try to get an intuitive feel for what they mean beyond my short explanations and why. The explanations are short on purpose, so you have something to think about. Feel free to post your thoughts (and questions) on the forum! Axiom of extensionality: a b[ c(c a c b) a = b]. This is our familiar definition of equality. We define two sets to be equal if they are subsets of each other. We saw this earlier in our discussion of language. Axiom of union: a b c d[(d c c a) d b]. 8

9 This says that if a is a family of sets, then b is a set containing all elements in any member of a. That is, b is the union of all sets in a. For a challenge, here is one more that I will not give any explanation to. Try to figure out what it means! Hint: think about subsets. a b c[( d(d c d a)) c b] Some of the other axioms are actually axiom schema meaning they are really an aggregate of an infinite number of axioms. One important such axiom schema is called the axiom schema of specification. It states that if we have a set a and a predicate P which has domain a, then there is a set b such that x b P (b). That is, we have a subset of a containing all elements that satisfy P. You can think of P as a property of some elements. If a is the integers, we could have P be the property of being even. Then b is the set of all even numbers. In order to keep everything within the realm of first-order logic, we need this to be an axiom schema since we cannot quantify over all predicates. These rules (and the others, which you are encouraged to look up) give us all possible sets. We can now, for example, realize the numbers as sets. We can just give special names to some of our sets. Question: Recall the definition of the natural numbers as sets in Section 2. Try to use the ZF axioms to build those! You will need the empty set, the Axiom of pairing (see Section 2), and the Axiom of union. You can write down that definition in steps, and each step should be justified by one of the above axioms. Now, we have a universe to work in. We can now explore our universe. This exploration could literally go on forever, so we will append this section to finite length in order to keep things efficient. 7 Axiom of Choice Many of you have probably heard of the Axiom of Choice. It is an independent statement from ZF, meaning the ZF axioms can neither prove nor disprove the statement (more discussion on independent statements shortly). The axiom of choice states that for any collection of sets, all of which are 9

10 nonempty, there is a way to pick one object out of each set. For example, if we had the set of all possible intervals of real numbers, the Axiom of choice says we can pick one number from each of them. This sounds intuitively true, but for huge sets with uncountably many elements (more on sizes of sets in other posts), it is a bit strange to think that we could actually do so. It involves choosing infinitely many objects from infinitely many sets. Using this axiom, we can prove a great deal of powerful theorems in many different areas of mathematics. One particularly fun example is proving the Banach-Tarski Paradox, which may show up in another post. Together with the Axiom of Choice, we call the entire set of axioms ZFC, which is what most of modern mathematics rests upon. Here is the statement of the Axiom of choice, using shorthand for several pieces. Try to intuitively understand this statement, and perhaps even try to rewrite it in full formality, avoiding the shorthand. a[ / a (f : a a) b a(f(b) b)]. This version is phrased as the existence of a choice function f, where f takes each member of the family a to the element we chose from it. To write this out completely, one would have to avoid the use of the union of a, the function f (and instead write f as a set, like I showed in Section 2), and avoid the use of the empty set. 8 Gödel Incompleteness We have already seen that the axiom of choice is not provable (or disprovable) by ZF axioms. Such a statement is said to be independent of ZF. If such a statement exists, then the theory is said to be incomplete. If you read the cardinality post, you will see that the continuum hypothesis is independent of ZFC. Another problematic type of statement for an axiom system has precisely the opposite problem: one can prove both the statement and its negation. That is, for some statement S, one can prove that S is true, and that S is true. If this is possible, then the theory is said to be inconsistent. As it turns out, no sufficiently powerful theory (system of axioms) can avoid both of these problems. By sufficiently powerful we mean, a system in which we can prove basic facts of arithmetic, such as the system of Peano axioms. This notion of sufficiently powerful can be made much more rigorous, but it is difficult to do so. For now, we can think of the theory of Peano arithmetic as our primary example of such a theory. The celebrated Gödel Incompleteness Theorem states that any such sufficiently powerful system of 10

11 axioms cannot be both consistent and complete. Gödel is known for several other important theorems with similar names, but this is perhaps the most famous of them. Proving these theorems involve sophisticated techniques. To prove this theorem, we assume consistency, and a special sentence which asserts its unprovability, called the Gödel Sentence, is constructed. The full details are beyond the scope of this post. Notice that this has some frightening consequences. This means that we can never have a perfect axiom system. In any theory, we can always find a contradiction, or find statements that are unprovable (and undisprovable ). This is a foundational problem in axiomatic logic and set theory. Studying advanced logic and set theory will reveal many fascinating independent statements. One can also use independent statements as assumptions to prove additional theorems, or prove statements which are close to the independent statement. This exploration becomes highly technical and complicated very quickly. 9 Concluding Remarks The foundational roots of mathematics have been explored extensively in the past two hundred years, and there are numerous views and formalisms justifying the transition from vague notions of truth up to the formal mathematics that we use today. The Zermalo-Fraenkel axioms form an extremely powerful theory in which to do mathematics, and as we have explored, one can formalize the idea of proving theorems in an axiomatic system very nicely. Here, we have seen just a taste of the thought that has gone into formalizing mathematics. The study of mathematical logic and set theory leads one into the depths of meta-mathematics, and is a study to which all mathematicians should be exposed. 11