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Georg Cantor, Kurt Gödel, and the Incompleteness of Mathematics

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Georg Cantor, Kurt Gödel, and the Incompleteness of Mathematics Georg Cantor, Kurt G¨odel,and the Incompleteness of Mathematics Ross Dempsey Department of Physics and Astronomy Johns Hopkins University JHU Splash, 2018 Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 1 / 27 Outline 1 Set Theory Na¨ıveSet Theory Infinite Sets Russell's Paradox 2 Formal Systems and G¨odel'sTheorem Formal Systems G¨odel'sTheorem 3 Insanity Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 2 / 27 What is a Set? 1 0 A set is a collection of objects, denoted S = 1; r; E; 1 1 The objects within a set are known as elements, and are denoted 1 2 S or 2 62 S If every element of T is an element of S, then T ⊆ S How many subsets does a set with n elements have? The empty set with no elements is denoted ?. ? ⊂ S for every set S Union of sets is denoted A [ B; intersection is A \ B Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 3 / 27 Category of Sets A function f : A ! B between two sets A and B specifies an element f(a) 2 B for every a 2 A How many functions are there from a set A with n elements to a set B with m elements? The category of sets consists of all the sets together with all set functions An isomorphism (or bijection) between two sets A and B consists of: Two functions, f : A ! B and g : B ! A Such that f ◦ g = 1A and g ◦ f = 1B Isomorphism partitions the sets into classes, called cardinal numbers Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 4 / 27 Cardinal Numbers Every natural number (0, 1, 2, :::) is a cardinal number What about infinite sets? Are they all isomorphic? N = f0; 1; 2;:::g Z = f:::; −2; −1; 0; 1; 2;:::g 2Z = f:::; −4; −2; 0; 2; 4;:::g Q = f1=2;p68=37; 4=5;:::g R = f1; e; 5;:::g Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 5 / 27 But try listing out all the real numbers f(n) in decimal form: f(0) = 1:18736582175 f(1) = 3:23904875843 f(2) = 6:85789342753 f(3) = 0:52118781334 Use the diagonal to construct a new number: 0:2482 ::: The new number does not equal f(n) for any n The cardinal of R is denoted c, and called the cardinal of the continuum Cardinal of the Continuum Georg Cantor proved that the natural numbers N are not isomorphic to the real numbers R The argument is called Cantor diagonalization. Take any function f : N ! R. Then: To be a bijection, f must \hit" every real number Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 6 / 27 Use the diagonal to construct a new number: 0:2482 ::: The new number does not equal f(n) for any n The cardinal of R is denoted c, and called the cardinal of the continuum Cardinal of the Continuum Georg Cantor proved that the natural numbers N are not isomorphic to the real numbers R The argument is called Cantor diagonalization. Take any function f : N ! R. Then: To be a bijection, f must \hit" every real number But try listing out all the real numbers f(n) in decimal form: f(0) = 1:18736582175 f(1) = 3:23904875843 f(2) = 6:85789342753 f(3) = 0:52118781334 Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 6 / 27 The new number does not equal f(n) for any n The cardinal of R is denoted c, and called the cardinal of the continuum Cardinal of the Continuum Georg Cantor proved that the natural numbers N are not isomorphic to the real numbers R The argument is called Cantor diagonalization. Take any function f : N ! R. Then: To be a bijection, f must \hit" every real number But try listing out all the real numbers f(n) in decimal form: f(0) = 1:18736582175 f(1) = 3:23904875843 f(2) = 6:85789342753 f(3) = 0:52118781334 Use the diagonal to construct a new number: 0:2482 ::: Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 6 / 27 The cardinal of R is denoted c, and called the cardinal of the continuum Cardinal of the Continuum Georg Cantor proved that the natural numbers N are not isomorphic to the real numbers R The argument is called Cantor diagonalization. Take any function f : N ! R. Then: To be a bijection, f must \hit" every real number But try listing out all the real numbers f(n) in decimal form: f(0) = 1:18736582175 f(1) = 3:23904875843 f(2) = 6:85789342753 f(3) = 0:52118781334 Use the diagonal to construct a new number: 0:2482 ::: The new number does not equal f(n) for any n Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 6 / 27 Cardinal of the Continuum Georg Cantor proved that the natural numbers N are not isomorphic to the real numbers R The argument is called Cantor diagonalization. Take any function f : N ! R. Then: To be a bijection, f must \hit" every real number But try listing out all the real numbers f(n) in decimal form: f(0) = 1:18736582175 f(1) = 3:23904875843 f(2) = 6:85789342753 f(3) = 0:52118781334 Use the diagonal to construct a new number: 0:2482 ::: The new number does not equal f(n) for any n The cardinal of R is denoted c, and called the cardinal of the continuum Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 6 / 27 Infinity of Infinities Cantor diagonalization can be used to prove a more general fact Let S be a set and 2S be its power set, the set of subsets of S Let f : S ! 2S be any set function. Then construct a subset T ⊂ S according to: If s 2 f(s), s 62 T If s 62 f(s), s 2 T Then T 6= f(s) for any s 2 S This means the cardinal of 2S is greater than the cardinal of S, for any S Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 7 / 27 Infinity Can't Be That Easy Let S = fsets not containing themselvesg = fx 2 Set j x 62 xg Does S contain itself? If S 2 S, then S 62 S If S 62 S, then S 2 S Put another way: a barber shaves every man in his town who does not shave themselves. Does the barber shave himself? This is Russell's paradox. It shows that na¨ıveset theory is inconsistent. Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 8 / 27 Zermelo-Frankel Set Theory Set theory is the foundation of all mathematics. It must be made consistent The Zermelo-Frankel (ZF) axioms fix Russell's paradox Key idea: build a universe of sets in stages: sets, sets containing sets, sets containing sets containing sets, ... Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 9 / 27 Axiom of Choice ZF set theory is often endowed with an additional axiom: the axiom of choice. It becomes ZFC set theory The axiom of choice says that, given a collection of sets, it is possible to choose a single element from each set Does this seem reasonable? Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 10 / 27 Well-ordering theorem: every set can be ordered in such a way that every subset has a least element Non-measurable sets: there exist subsets of the plane which cannot be assigned an area You can win an infinite hat game However, AC is also essential for many fundamental results in mathematics. So we keep it in. Axiom of Choice Infinity is a dangerous game An axiom should be judged by its consequences. Consequences of AC: Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 11 / 27 Non-measurable sets: there exist subsets of the plane which cannot be assigned an area You can win an infinite hat game However, AC is also essential for many fundamental results in mathematics. So we keep it in. Axiom of Choice Infinity is a dangerous game An axiom should be judged by its consequences. Consequences of AC: Well-ordering theorem: every set can be ordered in such a way that every subset has a least element Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 11 / 27 You can win an infinite hat game However, AC is also essential for many fundamental results in mathematics. So we keep it in. Axiom of Choice Infinity is a dangerous game An axiom should be judged by its consequences. Consequences of AC: Well-ordering theorem: every set can be ordered in such a way that every subset has a least element Non-measurable sets: there exist subsets of the plane which cannot be assigned an area Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 11 / 27 However, AC is also essential for many fundamental results in mathematics. So we keep it in. Axiom of Choice Infinity is a dangerous game An axiom should be judged by its consequences. Consequences of AC: Well-ordering theorem: every set can be ordered in such a way that every subset has a least element Non-measurable sets: there exist subsets of the plane which cannot be assigned an area You can win an infinite hat game Ross Dempsey (Johns Hopkins University) Cantor, G¨odel,and Incompleteness JHU Splash, 2018 11 / 27 Axiom of Choice Infinity is a dangerous game An axiom should be judged by its consequences.
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