Introduction to mathematical thinking Alexandru Buium Department of Mathematics and Statistics, University of New Mex- ico, Albuquerque, NM 87131, USA E-mail address: [email protected] This is a (drastically) simplified version of my book: Mathematics: a minimal introduction, CRC press, 2013. For a more complete treatment of the topics one may refer to the book. However one should be aware of the fact that there are some key conceptual differences between the present text and the book, especially when it comes to the material that pertains to logic (e.g., to witnesses, quantifier axioms, and the structure of theories). Contents Part 1. Logic 5 Chapter 1. Languages 7 Chapter 2. Metalanguage 15 Chapter 3. Syntax 21 Chapter 4. Tautologies 25 Chapter 5. Proofs 31 Chapter 6. Theories 39 Chapter 7. ZFC 49 Part 2. Set theory 53 Chapter 8. Sets 55 Chapter 9. Maps 59 Chapter 10. Relations 63 Chapter 11. Operations 69 Part 3. The discrete 75 Chapter 12. Integers 77 Chapter 13. Induction 81 Chapter 14. Rationals 85 Chapter 15. Combinatorics 87 Chapter 16. Sequences 89 Part 4. The continuum 93 Chapter 17. Reals 95 Chapter 18. Topology 97 Chapter 19. Imaginaries 101 3 4 CONTENTS Part 5. Algebra 103 Chapter 20. Arithmetic 105 Chapter 21. Groups 109 Chapter 22. Order 113 Chapter 23. Vectors 115 Chapter 24. Matrices 117 Chapter 25. Determinants 121 Chapter 26. Polynomials 125 Chapter 27. Congruences 129 Part 6. Geometry 133 Chapter 28. Lines 135 Chapter 29. Conics 139 Chapter 30. Cubics 141 Part 7. Analysis 145 Chapter 31. Limits 147 Chapter 32. Series 151 Chapter 33. Trigonometry 155 Chapter 34. Calculus 157 Part 1 Logic CHAPTER 1 Languages Mathematics is a theory called set theory. Theories are (growing) sequences of sentences. The formation of sentences and theories is governed by (general) logic (not to be confused with mathematical logic which is part of mathematics). Logic starts with the analysis/construction of language. Example 1.1. (Logical analysis of language) We will introduce here two lan- guages, English and Formal, and we will analyze their interconnections. Let us start with a discussion of English. The English language is the collection LEng of all English words (plus separators such as parentheses, commas, etc.). We treat words as individual symbols (and ignore the fact that they are made out of letters). Sometimes we admit as symbols certain groups of words. One can use words to create strings of words such as 0)\for all not Socrates man if" The above string is considered \syntactically incorrect." The sentences in the English language are the strings of symbols that are \syntactically correct" (in a sense to be made precise later). Here are some examples of sentences in this language: 1) \Socrates is a man" 2) \Caesar killed Brutus" 3) \The killer of Caesar is Brutus" 4) \Brutus killed Caesar and Socrates is a man" 5) \Brutus is not a man or Caesar is a killer" 6) \If Brutus killed Caesar then Brutus is a killer" 7) \Brutus did not kill Caesar" 8) \A man killed Caesar" 9) \If a man killed another man then the first man is a killer" 10) \A man is a killer if and only if that man killed another man" In order to separate sentences from a surrounding text we put them between quotation marks (and sometimes we write them in italics). So quotation marks do not belong to the language but rather they lie outside the language; they belong to metalanguage, as we shall explain. Checking syntax presupposes a partitioning of LEng into various categories of words; no word should appear in principle in two different categories, but this requirement is often violated in practice (which may lead to different readings of the same text). Here are the categories: • variables: \something,..." • constants: \Socrates, Brutus, Caesar,..." • functional symbols: \the killer of,..." 7 8 1. LANGUAGES • predicates: \is a man, is a killer, killed,..." • connectives: \and, or, not, if...then, if and only if" • quantifiers: \for all, there exists" • equality: \is, equals" • separators: parentheses \(,)" and comma \," The above categories are referred to as logical categories. (They are quite different from, although related to, the grammatical categories of nouns, verbs, etc. In general objects are named by constants or variables. (So constants and variables roughly correspond to proper nouns.) Constants are names for specific objects while variables are names for non-specific (generic) objects. The article \the" generally indicates a constant; the article \a" generally indicates that a quantifier is implicitly assumed. Predicates say/affirm something about one or several objects; if they say/affirm something about one, two, three objects, etc., they are unary, binary, ternary, etc. (So roughly unary predicates correspond to intransitive verbs; binary predicates correspond to transitive verbs.) \Killed" is a binary predicate; \is a killer" is a unary predicate. Functional symbols have objects as arguments but do not say/affirm anything about them; all they do is refer to (or name, or specify, or point towards) something that could itself be an object. (Functional symbols are sometimes referred to as functional predicates but we will not refer to them as predicates here; this avoids confusion with predicates.) Again they can be unary, binary, ternary, etc., depending on the number of arguments. \The father of" is a unary functional symbol. \The son of ... and ..." is a binary functional symbol (where the two arguments stand for the mather and the father and we assume for simplicity that any two parents have a unique son.) Connectives connect/combine sentences into longer sentences; they can be unary (if they are added to one sentence changing it into another sentence, binary if they combine two sentences into one longer sentence, ternary, etc.). Quantifiers specify quantity and are always followed by variables. Separators separate various parts of the text from various other parts. In order to analyze a sentence using the logical categories above one first looks for the connectives and one splits the sentence into simpler sentences; alternatively sentences may start with quantifiers followed by variables followed by simpler sen- tences. In any case, once one identifies simpler sentences, one proceeds by iden- tifying, in each of them, the constants, variables, and functional symbols applied to them (these are the objects that one is talking about), and finally one identifies the functional symbols (which say something about the objects). The above type of analysis (called logical analysis) is quite different from the grammatical analysis based on the grammatical categories of nouns, verbs, etc. A concise way of understanding the logical analysis of English sentences as above is to create another language LF or (let us call it Formal) consisting of the following symbols: • variables: \x; y; :::" • constants: \S; B; C; :::" • functional symbols: \#; :::" • predicates: \m; k; y" • connectives: \^; _; :; !; $" • quantifiers: \8; 9" • equality: \=" 1. LANGUAGES 9 • separators: parentheses \(; )" and comma \;" Furthermore let us introduce a rule (called translation) that attaches to each symbol in Formal a symbol in English as follows: \x; y" are translated as \something, a thing, an entity,..." \S; B; C" are translated as \Socrates, Brutus, Caesar" \#; :::" are translated as \the killer of,..." \m; k; y" are translated as \is a man, is a killer, killed,..." \^; _; :; !; $" are translated as \and, or, not, if...then, if and only if" \8; 9" are translated as \for all, there exists" \=" is translated as \is" or \is a equal to" Then the English sentences 1-10 are translations of the following Formal sentences; equivalently the following Formal sentences are translations (called formalizations) of the corresponding English sentences: 1') \m(S)" 2') \C y B" 3') \# (C) = B" 4') \(B y C) ^ m(S)" 5') \(:(m(B))) _ k(C)" 6') \(B y C) ! (k(B))" 7') \:(B y C)" 8') \9x(x y C)" 9') \8x((m(x) ^ (9y(m(y) ^ :(x = y) ^ (x y y))) ! k(x))" 10') \8x(k(x) $ (m(x) ^ (9y(m(y) ^ :(x = y) ^ (x y y))))" In the above formalizations one first replaces 8), 9), 10) by 8") \There exists something such that that something is a man and that some- thing killed Caesar" 9") \For any x if x is man and there exists a y such that y is a man, y is different from x and x killed y then x is a killer" 10") \For any x one has that x is a killer if and only if x is a man and there exists y such that y is a man, x is not y, and x killed y" Note that the word \exists" which has the form of a predicate is considered instead as part of a quantifier. Sentences like \philosophers exist" and \philosophers are human" have a totally different logical structure. Indeed \philosophers exist" should be read as \there exists something such that that something is a philosopher" while \philosophers are human" should be read as \for all x if x is a philosopher then s is a human." The fact that \exist" should not be viewed as a predicate was recognized already by Kant, in particular in his criticism of the \ontological argument." On the other hand the verb to be (as in \is, are,...") can be: i) part of a predicate (as in \is a man"), ii) part of equality (as in \is equal, is the same as"), iii) part of a quantifier (as is \there is", an equivalent translation of 9). All of our discussion of English and Formal above is itself expressed in yet another language which needs to be distinguished from English itself and which we shall call Metalanguage. We will discuss Metalanguage in detail in the next chapter (where some languages will be declared object languages and others will be declared 10 1.